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U.  S.  GEOLOGICAL  SURVEY 


PROFESSIONAL  PAPER  86     PLATE  I 


APPARATUS    ON    CAMPUS    OF    UNIVERSITY    OF    CALIFORNIA. 


DEPARTMENT  OF  THE  INTERIOR 
UNITED  STATES  GEOLOGICAL  SURVEY 

GEORGE  OTIS  SMITH,  DIBECTOR 


By  CHARLES  QILMAN  JTTDg 


Referred  to 


Attended  to  by 
"        "  whe7 


file  wrier 


/<•.„,,.  ; 


PROFESSIONAL  PAPER  86 


THE 


TRANSPORTATION  OF  DEBRIS  BY  RUNNING  WATER 


BY 

GROVE   KARL  GILBERT 

t< 

BASED  ON  EXPERIMENTS  MADE  WITH  THE  ASSISTANCE  OF 

EDWARD  CHARLES  MURPHY 


WASHINGTON 

GOVERNMENT    PRINTING    OFFICE 

1914 


1HGINEERING  LIBRARt 


o" 


CONTENTS. 


r>  v, 


Preface 

Abstract 

Notation 

CHAPTER  I.  The  observations 

Introduction 

General  classification 

Stream  traction 

Outline  of  course  of  experimentation 

Scope  of  experiments 

Accessory  studies 

Flume  traction 

Apparatus  and  material 

Experiment  troughs 

Water  supply 

The  water  circuit 

Discharge 

Sand  feed 

Sand  arrester 

Settling  tank 

Gage  for  depth  measurement 

Level  for  slope  measurement 

Pitot-Darcy  gage 

Sand  and  gravel 

Methods  of  experimentation 

General  procedure  for  a  single  experiment. 

Details  of  procedure 

Width  of  channel 

Discharge 

The  feeding  of  sand 

The  collection  of  sand 

Determination  of  load 

Determination  of  slope 

Contractor 

Measurement  of  depth 

Measurement  of  velocity 

Modes  of  transportation 

Movement  of  individual  particles 

Rolling . 

Saltation 

Collective  movement 

Units 

Terms 

Load 

Capacity .* 

Competence 

Discharge 

Slope 

Size  of  debris 

Form  ratio 

Duty  and  efficiency 

Symbols 

Table  of  observations  on  stream  traction.  . 


Tage. 

9      CHAPTER  II.  Adjustment  of  observations 55 

10             Observations  on  capacity  and  slope 55 

13                    The  observational  series 55 

15                     Errors 56 

15                     Intake  influences 56 

15                     Outfall  influences 56 

15                     Changes  in  apparatus 57 

17                     Rhythm 58 

17  Slopes  of  debris  and  water  surface 59 

18  The  logarithmic  plots 59 

18  Selection  of  an  interpolation  formula 60 

19  The  constant  a  and  competent  slope 65 

19                     Interpolation 72 

'19                     Precision 73 

19  Duty 74 

20  Efficiency 75 

20             Observations  on  depth 87 

20  Mode  of  adjustment 87 

21  Precision 94 

21                     Mean  velocity 94 

21                      Form  ratio 94 

21                    Graphic  computation 94 

21  CHAPTER  III.  Relation  of  capacity  to  slope 96 

22  Introduction 96 

22              In  channels  of  fixed  width 96 

22                    The  conditions 96 

22                     The  sigma  formula 96 

22  The  power  function  and  the  index  of  rela- 

23  tive  variation 97 

24  The  synthetic  index 99 

24  Application  to  the  sigma  function 99 

25  Variation  of  the  index 104 

25                    Formulation  with  constant  coefficient 109 

25  Effect  of  changing  the  unit  of  slope 112 

26  Precision 113 

26  Evidence  from  experiments  with  mixed 

26                        debris 113 

26                     Relation  of  index  to  mode  of  traction 115 

26             In  channels  of  similar  section 116 

30                    The  conditions 116 

34  Sigma  and  the  index 117 

35  The  synthetic  index 119 

35                     Summary 120 

35             Review 120 

35             Duty  and  efficiency 121 

35      CHAPTER  IV.  Relation  of  capacity  to  form  ratio 124 

35             Introduction 124 

35             Selection  of  a  formula 124 

35  Maximum 124 

36  Capacity  and  width 125 

36                    Capacity  and  depth -. .  127 

36                    Capacity  and  form  ratio 128 

3 


CONTENTS. 


Page. 
CHAPTER  IV — Continued. 

Discussion  of  experimental  data 130 

Scope  and  method  of  discussion 130 

Sensitiveness  and  the  index  of  relative  vari- 
ation   130 

Control  of  constants  by  slope 131 

Control  of  constants  by  discharge 132 

Control  of  constants  by  fineness 133 

Special  group  of  observations 134 

Summary  as  to  control  by  conditions 134 

The  optimum  form  ratio 135 

Summary 136 

CHAPTER  V.  Relation  of  capacity  to  discharge 137 

Formulation  and  reduction 137 

Measures  of  precision  and  their  interpretation . .  142 

Control  of  relative  variation  by  conditions 143 

Duty  and  efficiency 144 

Comparison  of  the  controlsof  discharge  and  slope .  145 

Controls  of  capacity 145 

Controls  of  duty 147 

Controls  of  efficiency 148 

Summary 149 

CHAPTER  VI.  Relation  of  capacity  to  fineness  of 

debris 150 

Formulation 150 

Precision 151 

Variation  of  the  constant  <f> 153 

Index  of  relative  variation 153 

Duty  and  efficiency ' 154 

Summary 154 

CHAPTER  VII.  Relation  of  capacity  to  velocity 155 

Preliminary  considerations 155 

The  synthetic  index  when  discharge  is  constant.  157 

Mean  velocity  versus  slope 158 

The  synthetic  index  when  slope  is  constant —  159 

The  synthetic  index  when  depth  is  constant. . .  160 

The  three  indexes 160 

Relative  sensitiveness  to  controls 162 

Competent  velocity 162 

CHAPTER  VIII.  Relation  of  capacity  to  depth 164 

Introduction 164 

When  discharge  is  constant 164 

When  slope  is  constant 164 

Depth  versus  discharge 165 

When  velocity  is  constant 165 

The  three  conditions  compared 166 

Comparison  of  controls  by  slope,  discharge,  and 

mean  velocity 167 

CHAPTER  IX.  Experiments  with  mixed  grades 169 

Adjustment  and  notation 169 

Mixtures  of  two  grades 172 

Control  by  slope  and  discharge 175 

Mixtures  of  more  than  two  grades 176 

A  natural  grade 177 

Causes  of  superior  mobility  of  mixtures 178 

Voids 179 

Fineness 180 

Relation  of  capacity  to   fineness,  for  natural 

grades 180 

Definition  and  measurement  of  mean  fineness..  182 

Summary 184 


Page. 

CHAPTER  X.  Review  of  controls  of  capacity 186 

Introduction 186 

Formulation  based  on  competence 186 

The  form-ratio  factor 190 

Duty  and  efficiency 192 

The  formula  of  Lechalas 193 

The  formula 193 

Discussion 194 

CHAPTER   XI.    Experiments  with   crooked   chan- 
nels    196 

Experiments 196 

Slope  determinations 196 

Forms  and  slopes 197 

Features  caused  by  curvature 198 

CHAPTER  XII.  Flume  traction 199 

The  observations 199 

Scope 199 

Grades  of  debris 199 

Apparatus  and  methods 199 

Processes  of  flume  traction 200 

Movement  of  individual  particles.  .  .  .  200 

Collective  movement 201 

Table  of  observations 202 

Adjustment  of  observations 203 

Formulation 203 

Precision 206 

Discussion 206 

Capacity  and  channel  bed 206 

Capacity  and  slope 208 

Capacity  and  discharge 209 

Capacity  and  fineness 210 

Mixtures 212 

Capacity  and  form  ratio 213 

Trough  of  semicircular  cross  section 214 

Summary 215 

Competence 215 

W'ork  of  Overstrom  and  Blue 216 

CHAPTER  XIII.  Application  to  natural  streams.  ...  219 

Introduction 219 

Features  distinguishing  natural  streams 219 

Kinds  of  streams 219 

Features    connected    with    curvature    of 

channel 220 

Features  connected  with  diversity  of  dis- 
charge   221 

Sections  of  channel 222 

The  suspended  load 223 

The  two  loads 230 

Availability  of  laboratory  results 233 

The  slope  factor 233 

The  discharge  factor 233 

The  fineness  factor 235 

The  form-ratio  factor 236 

The  four  factors  collectively 236 

The  hypothesis  of  similar  streams 236 

Summary 240 

Conclusion 240 

CHAPTER  XIV.  Problems  associated  with  rhythm..  241 

Rhythm  in  stream  transportation 241 

Rhythm  in  the  flow  of  water 242 


CONTENTS. 


Page. 

CHAPTER  XIV — Continued. 

The  vertical  velocity  curve 244 

The  moving  field 249 

APPENDIX  A.  The  Pitot-Darcy  gage 251 

Scope  of  appendix 251 

Form  of  instrument 251 

Rating  methods 252 

Rating  formula 253 


Page. 
APPENDIX  B.  The  discharge-measuring  gate  and  its 

rating 257 

The  gate 257 

Plan  of  rating 258 

Calibration  of  measuring  reservoir 258 

The  observations 259 

INDEX..  261 


TABLES. 


J'age. 
TABLE    1.  Grades  of  de'bris 21 

2.  Gate   readings  and    corresponding   dis- 

charges          23 

3.  Data  connected  with  changes  in  mode 

of  transportation 33 

4.  Observations  on  load,  slope,  and  depth..        38 

5.  Values  of  capacity  generalized  from  Table 

4(G) 61 

6.  Data  for  the  construction  of  curves  in 

figures  20  and  21 62 

7.  Values  of  a  in  C=6,(S-<T)B 66 

8.  Values  of  a,  arranged  to  show  variation 

in  relation  to  fineness  of  debris 68 

9.  Experimental  data  on  competent  slope..        69 

10.  Experimental   data  on   competent  dis- 

charge          70 

11.  Values  of  a,  as  adjusted  for  use  in  inter- 

polation equations 72 

12.  Adjusted  values  of  capacity,  duty,  and 

efficiency 


75 
89 


13.  Values  of  nl  in  d=— — 

14.  Adjusted  values  of  depth  of  current,  with 

values  of  mean  velocity  and  form  ratio .  89 

15.  Values  of  i\  in  relation  to  slope 100 

16.  Valuesof  j,  in  C=cS>i 109 

17.  Values  of  i,  for  mixtures  of  two  or  more 

grades 114 

18.  Comparison  of  values  of  ^  for  mixtures 

and  their  components 115 

19.  Values  of  il  associated  with  the  smooth 

mode  of  traction 115 

20.  Relations  of  il  to  discharge,  de'bris  grade, 

and  channel  width 116 

21.  Selected  data  showing  the  relation  of  ca- 

pacity to  slope  when  the  form  ratio  is 
constant 117 

22.  Values  of  a  corresponding  to  data    in 

Table  21 118 

23.  Values  of  synthetic  index  under  condi- 

tion that  R  is  constant,  and  under  con- 
dition that  w  is  constant 119 

23a.  Comparison    of    parameters    in    C=5, 

(S-<j)»and  E=B1(S-a)nn 123 

24.  Relation   of   capacity   to   width    when 

slope  and  depth  are  constant 126 

25.  Values  of  capacity  and  depth  for  grade 

(C),  width  0.66  foot,  and  slope  1.0  per 
cent .  .  128 


TABLE  26.  Values  of  ml,  in  Coc  d1*1,  when  slope  is 
constant 

27.  Quantities  illustrating  the  influence  of 

slope  on  the  relation  of  capacity  to  form 
ratio 

28.  Quantities  illustrating  the  influence  of 

discharge  on  the  relation  of  capacity 
to  form  ratio 

29.  Quantities  illustrating  the  influence  of 

fineness  on  the  relation  of  capacity  to 
form  ratio 

30.  Quantities  illustrating  the  influence  of 

slope  and  discharge  on  the  relation  of 
capacity  to  form  ratio 

31.  Estimated  ratios  of  depth  to  width  to 

enable  a  given  discharge,  on  a  given 
slope,  to  transport  its  maximum  load. 

32.  Data  on  the  relation  of  capacity  to  dis- 

charge   

33.  Values  of   K,   computed    from   data   of 

Table  32 

34.  Adjusted  system  of  values  of  K 

35.  Values  of  i3,  arranged  to  show  variation 

in  relation  to  fineness 

36.  Comparison  of  il  with  ig 

37.  Variations  of  i,  arid  ig  in  relation  to  dis- 

charge   

38.  Variations  of  i,  and  i3  in  relation  to  width . 

39.  Variations  of  il  andt'3  in  relation  to  fine- 

ness   

40.  Variations  of  f,  and  i,3  in  relation  toslope. 

41.  Variationsof  the  ratio  -^-  . 


42.  Variations  of  the  ratio  -J 


1 


Page. 

128 

132 
133 
133 
134 

135 
137 

140 
141 

144 

145 

146 
146 

147 
147 

148 
148 


43 


Values  of  capacity,  arranged  to  illustrate 
the  relation  of  capacity  to  grades  of 
de'bris 150 

Data  connected  with  the  plots  in  fig- 
ure 50  151 

45.  Values  of  Tt,  compared  with  slope,  dis- 
charge, and  width 154 

Values  of  Ivq 157 

Means  based  on  Table  46,  illustrating  the 
control  of  /FQ  by  discharge,  fineness, 

and  width 158 

48.  Synthetic  indexes,  comparing  the  con- 
trol of  capacity  by  mean  velocity  with 
the  control  by  slope 158 


44 


46 

47 


CONTENTS. 


Page. 
TABLE  49.  Values  of  It>, 159 

50.  Means   based  on  Table  49,  illustrating 

the  control  of  Iva  by  slope,  fineness, 

and  width 159 

51.  Values  of  In 160 

52.  Means  based  on  Table  51,  illustrating  the 

control  of  Ira  by  fineness  and  width. .       160 

53.  Comparison  of  synthetic  indexes  for  ca- 

pacity and  mean  velocity,  under  the 
several  conditions  of  constant  discharge, 
constant  depth,  and  constant  slope 161 

54.  Values  of  las 164 

55.  Means  based  on  Table  54,  illustrating 

the  control  of  Ids  by  slope,  fineness, 

and  width 165 

56.  Synthetic  indexes  comparing  the  control 

of  capacity  by  depth  with  control  by 
discharge 165 

57.  Values  of  I,iv 165 

58.  Means  based  on  Table  57,  illustrating  the 

control  of  Irir  by  mean  velocity,  fine- 
ness, and  width 166 

59.  Comparison  of  synthetic  indexes  for  ca- 

pacity and  depth  under  the  several 
conditions  of  constant  discharge,  con- 
stant elope,  and  constant  mean  velocity  167 

60.  Adjusted  values  of  capacity  for  mixtures 

of  two  or  more  grades  of  d<5bris 169 

61.  Capacities  for  traction,  with  varied  mix- 

tures of  two  grades 172 

62.  Percentages  of  voids  in  certain  mixed 

grades  of  debris 179 

63.  Fineness  of  mixed  grades  and  their  com- 

ponents         180 

63a.  Data  on  subaqueous  dunes  of  the  Loire.       194 

64.  Comparison  of  slopes  required  forstraight 

and  crooked  channels 197 

65.  Grades  of  debris 199 


Pag,-. 

TABLE  66.  Relative  speeds  of  coarse  and  fine  de- 
bris in  flume  traction 200 

67.  Observations  on  flume  traction,  showing 

the  relation  of  load  to  slope  and  other 
conditions 202 

68.  Values  of  capacity  for  flume  traction,  ad- 

justed in  relation  to  slope 204 

69.  Comparison  of  capacities  associated  with 

different  characters  of  channel  bed 206 

70.  Values  of  /,  for  flume  traction 208 

71.  Comparison  of  values  of  7,  for  flume  trac- 

tion   with    corresponding    values    for 
stream  traction 209 

72.  Data  illustrating  the  relation  of  capacity 

for  flume  traction  to  discharge 209 

73.  Values  of  capacity  for  flume   traction, 

illustrating  the  control  of  capacity  by 
fineness 210 

74.  Capacities   for   mixed   grades  and  their 

components 212 

75.  Depths    and    form    ratios  of    unloaded 

streams 213 

76.  Capacities  for  flume  traction  in  troughs 

of  different  widths 213 

77.  Data  on  flume  traction  in  a  semicylindric 

trough 214 

78.  Observations  by  Blackwell  on  velocity 

competent  for  traction 216 

79.  Values  of  n  in  C  oc  Vm"  based  on  Blue's 

experiments 218 

80.  Velocities  of  streams  with  and  without 

fractional  loads 230 

81.  Ratio  of  the  suction  at  one  opening  of  the 

Pitot-Darcy  gage  to  the  pressure  at  the 

other 254 

82.  Values  of  K  and  u in  77— 7/,=A'K" 254 

83.  Values  of  A  in  V=  A^II^II, 255 


ILLUSTRATIONS. 


Pago. 
PLATE  I.  Apparatus  on  campus  of  the  University 

of  California Frontispiece. 

II.  De'bris  used  in  experiments 22 

III.  Rough  surfaces  used  in  experiments  on 

flume  traction 200 

FIGURE    1.  Diagrammatic  view  of  shorter  experi- 
ment trough 19 

2.  Diagram  of  water  circuit 20 

3.  The  contractor 25 

4.  Diagrammatic  view  of  part  of  experi- 

ment trough  with  glass  panels  and 
sliding  screen 27 

5.  Appearance  of  the  zone  of  saltation 27 

6.  The  beginning  of  a  leap,  in  saltation.  .         28 

7.  Diagram   of   accelerations   affecting   a 

saltatory  grain 28 

8.  Theoretic  trajectory  of  a  saltatory  par- 

ticle          29 

9.  Ideal  transverse  section  of  zone  of  sal- 

tation at  side  of  experiment  trough. .        30 


FIGURE  10.  Longitudinal   section   illustrating   the 
dune  mode  of  traction 

11.  Longitudinal    section    illustrating    the 

antidune  mode  of  traction 

12.  Profiles  of  water  surface,  automatically 

recorded,  showing  undulations  asso- 
ciated with  the  antidune  mode  of 
traction 

Plot  of  a  single  series  of  observations  of 
capacity  and  elope 

Logarithmic  plot  of  a  series  of  observa- 
tions on  capacity  and  slope 

Diagrammatic  longitudinal  section  of 
outfall  end  of  experiment  trough, 
illustrating  influence  of  sand  arrester 
on  water  slope 

Diagrammatic  longitudinal  section  of 
debris  bed  and  stream,  in  a  long 
trough , 


Page. 
;j  1 


33 


13. 


14. 


16. 


56 


57 


57 


CONTENTS. 


FIGURE  17.  Diagrammatic  longitudinal  section  of 
outfall  end  of  trough,  illustrating 
influence  of  contractor 

18.  Profiles   of   channel   bed,    illustrating 

fractional  rhythms  associated  with 
dunes  of  greater  magnitude 

19.  Logarithmic  graph  of  C=f(S),  for  grade 

(G),  «-=0.66  foot,  Q=0.734  ft.3/aec... 

20.  Extrapolated  curves  of  C=f(S)  for  vari- 

ous tentative  equations  of  interpola- 
tion and  for  slopes  greater  than  2.1 
per  cent 

21 .  Extrapolated  curves  of  C=/(S)  for  vari- 

ous tentative  equations  of  interpola- 
tion and  for  slopes  less  than  0.8  per 
cent 

22.  The  relation  of  a  in  C=f(S-a)  to  the 

curvature  of  the  logarithmic  graph. . 

23.  Diagrammatic    sections    of    laboratory 

troughs,  illustrating  relation  of  cur- 
rent depth  to  trough  width 

24.  Ideal  curve  of  competent  slope  in  rela- 

tion to  width  of  trough 

25.  Logarithmic  plot  of  competent  slope  in 

relation  to  fineness  of  debris 

26.  Illustration  of  the  method  used  to  ad- 

just values  of  capacity  in  relation  to 
slope  by  means  of  a  logarithmic  plot 
of  observed  values  of  capacity  in  re- 
lation to  slope  minus  a 

27.  Observations  of  depth  of  current  in  re- 

lation to  slope 

28.  Logarithmic  computation  sheet,   com- 

bining relations  of  capacity,  mean 
velocity,  form  ratio,  and  slope 

29.  Logarithmic  locus  of  the  exponential 

equation 

30.  Locus  of  log  .y=/(log  x),  illustrating  the 

nature  of  the  index  of  relative  varia- 
tion  

31.  Variations  of  i,  in  relation  to  slope 

32.  Variations  of  t,  in  relation  to  width  of 

channel 

33.  Variations  of  i,  in  relation  to  discharge. 

34.  Variations  of  i,  in  relation  to  fineness  of 

debris 

Variations  of  exponents  i, ,  j, .  and  k  in 
relation  to  slope 


35. 
36. 


form  mode  of  traction  and  uniform 
slope 

37.  Curves    of    C=f(S)    for    three    trough 

widths 

38.  Illustration  of  the  relation  of  capacity 

to  width  of  channel  and  to  form  ratio, 
when  slope  and  discharge  are  constant 

39.  Cross  sections  of  stream  channels 

40.  Capacity   for   traction   in    relation   to 

width  of  channel,  when  depth  and 
slope  are  constant 

41.  Plot  of  equation  (55) 

42.  Logarithmic  plots  of 


Page. 
57 

58 
61 

62 

63 
65 

67 
67 
69 

72 
87 

95 
98 

98 
105 

106 

107 

108 
113 

116 
118 


124 
125 


127 
129 

130 


Page. 
FIGURE  43.  Relation  of  capacity,  C,  to  form  ratio, 

R.  The  variation  of  the  function 
C=b2(l+nR)Rm  with  slope 131 

44.  Relation  of  capacity,  C,  to  form  ratio, 

R.  The  variation  of  the  function 
0=62(1-0-^)^™  with  discharge 132 

45.  Relation  of  capacity,  C,  to  form  ratio, 

R.  The  variation  of  the  function 
C=b,(l-aR)Rm  with  fineness  of  de- 
bris 133 

46.  Relation  of  capacity,  C,  to  form  ratio, 

R.  The  variation  of  the  function 
C=b,(l-aR)Rm  with  slope  and  dis- 
charge   134 

47.  Logarithmic   plots   of   the   relation   of 

capacity  to  discharge 139 

48.  Ideal  cross  section  of  a  stream  in  the  ex- 

periment trough,  illustrating  the  rela- 
tion of  competent  discharge  to  width .  140 

49.  Logarithmic  plot  of  K=f(D) 141 

50.  Logarithmic  plots  of  capacity  for  trac- 

tion in  relation  to  fineness  of  de'bris. .       150 

51.  Average  departures  of  original  values 

of  capacity  from  system  of  values 
readjusted  in  relation  to  fineness  of 
debris 152 

52.  Vertical  velocity  curve,  drawn  to  illus- 

trate its  theoretic  character  near  the 
stream's  bed 155 

53.  Ideal  profile  'of  a  stream  bed  composed 

of  debris  grains 155 

54.  Ideal  curves  of  velocity  in  relation  to 

depth,  illustrating  their  relation  to 

the  zone  of  saltation 161 

55.  Tractional  capacity  for  mixed  de'bris, 

in  relation  to  proportions  of  compo- 
nent grades,  with  associated  curves  of 
fineness  and  of  percentages  of  voids. .  173 

56.  Tractional  capacities  of  components  of 

mixed  grades  in  relation  to  the 
percentages  of  the  components  in  the 
mixtures 174 

57.  Curves  of  capacity  in  relation  to  slope 

for  grade  (A),  grade  (G),  and  mixtures 

of  those  grades 175 

58.  Curves  of  capacity  in  relation  to  slope 

for  grade  (B),  grade  (F),  and  mixtures 

of  those  grades 175 

59.  Curves  of  capacity  in  relation  to  slope 

for  grade  (C),  grade  (E),  and  mixtures 

of  those  grades 176 

60.  Curves  of  capacity  in  relation  to  slope 

for  a  mixture  of  five  grades,  (C,  D,  E, 
F,  G).  Comparison  of  mixture  curves 
for  three  discharges  and  of  mixture 
curve  with  curves  for  component 
grades 176 

61.  Capacity-slope  curves  for  related  mix- 

tures and  capacity-discharge  curves 

for  mixture  and  component  grades. .       177 

62.  Capacity-slope    curves    for   a    natural 

grade  of  debris,  compared  with  curves 

for  sieve-separated  grades 178 


CONTENTS. 


Page. 

FIGURE  63.  Curves  showing  the  relations  of  various 
quantities  to  the  proportions  of  fine 
and  coarse  components  in  a  mixture  of 
two  grades  of  debris,  (C)  and  (G) 179 

64.  Logarithmic  plots  of  capacity  in  relation 

to  linear  fineness,  for  related  mixtures 

of  debris 181 

65.  Curve  illustrating  the  range  and  dis- 

tribution of  finenesses  in  natural  and 
artificial  grades  of  debris 181 

66.  Typical  curves  illustrating  the  distri- 

bution of  the  sensitiveness  of  capacity 
for  traction  to  various  controlling  con- 
ditions   191 

67.  Plans  of  troughs  used  in  experiments  to 

show  the  influence  of  bends  on  trac- 
tion    196 

68.  Contoured  plat  of  stream  bed,  as  shaped 

by  a  current 198 

69.  Curves  illustrating  the  relation  of  ca- 

pacity for  flume  traction  to  fineness 

of  debris 211 

70.  Diagram  of  forces 224 

71.  Interference    by    suspended    particle 

with  freedom  of  shearing 226 

72.  Suggested  apparatus  for  automatic  feed 

of  debris 241 

73.  Modification  of  vertical  velocity  curve 

by  approach  to  outfall 244 

74.  Modification  of  vertical  velocity  curve 

by  approach  to  contracted  outfall . .       245 

75.  Plan  of  experiment  trough  with  local 

contraction 245 

76.  Profile  of  water  surface  in  trough  shown 

in  figure  75 245 

77.  Modification  of  vertical  velocity  curve 

by  local  contraction  of  channel 245 


Page. 

FIGURE  78.  Modification  of  vertical  velocity  curve 
when  mean  velocity  is  increased  by 
change  of  slope 245 

79.  Modification  of  vertical  velocity  curve 

when  mean  velocity  is  increased  by 
change  of  discharge 246 

80.  Modification  of  vertical  velocity  curve 

by  changes  in  the  roughness  of  the 
channel  bed 246 

81.  Modification  of  vertical  velocity  curve 

by  addition  of  load  to  stream,  with 
corresponding  increase  of  slope 246 

82.  Modification  of  vertical  velocity  curve 

by  addition  and  progressive  increase 

of  load 246 

83.  Ideal  longitudinal  section  of  a  stream, 

illustrating  hypothesis  to  account 
for  the  subsurface  position  of  the  level 
of  maximum  velocity 248 

84.  Diagrammatic  plan  of  suggested  mov- 

ing-field apparatus 250 

85.  Longitudinal  section  of  lower  end  of 

receiver  of  Pitot-Darcy  gage  No.  3, 
with  transverse  sections  at  three 
points 251 

86.  Cross  section  of  prism  of  water  in  trough, 

showing  positions  given  to  gage  open- 
ing in  various  ratings 255 

87.  Graphic  table  for  interpolating  values  of 

A  in.V=A  TJH—H!,  for  observations 
made  with  gage  3b  in  different  parts 
of  a  stream 256 

88.  Arrangement   of  apparatus   connected 

with  the  rating  of  the  discharge- 
measuring  gate 257 

89.  Elevation  and  sections  of  gate  for  the 

measurement  of  discharge 258 


PREFACE. 


Thirty-five  years  ago  the  writer  made  a 
study  of  the  work  of  streams  in  shaping  the 
face  of  the  land.  The  study  included  a 
qualitative  and  partly  deductive  investigation 
of  the  laws  of  transportation  of  de'bris  by 
running  water;  and  the  limitations  of  such 
methods  inspired  a  desire  for  quantitative 
data,  such  as  could  be  obtained  only  by 
experimentation  with  determinate  conditions. 
The  gratification  of  this  desire  was  long  de- 
ferred, but  opportunity  for  experimentation 
finally  came  in  connection  with  an  investigation 
of  problems  occasioned  by  the  overloading  of 
certain  California  rivers  with  waste  from 
hydraulic  mines.  The  physical  factors  of 
those  problems  involve  the  transporting  capac- 
ity of  streams  as  controlled  by  various  condi- 
tions. The  experiments  described  in  this 
report  were  thus  instigated  by  the  common 
needs  of  physiographic  geology  and  hydraulic 
engineering. 

A  laboratory  was  established  at  Berkeley, 
Cal.,  and  the  investigation  became  the  guest 
of  the  University  of  California,  to  which  it  is 
indebted  not  only  for  space,  within  doors  and 
without,  but  for  facilities  of  many  kinds  most 
generously  contributed. 

Almost  from  the  beginning  Mr.  E.  C.  Murphy 
has  been  associated  with  me  in  the  investigation 
and  has  had  direct  charge  of  the  experiments. 
Before  the  completion  of  the  investigation  I 
was  compelled  by  ill  health  to  withdraw  from 
it,  and  Mr.  Murphy  not  only  made  the  remain- 
ing series  of  experiments,  so  far  as  had  been 
definitely  planned,  but  prepared  a  report. 
This  report  did  not  incluae  a  full  discussion  of 
the  results  but  was  of  a  preliminary  nature, 
it  being  hoped  that  the  work  might  be  con- 
tinued, with  enlargement  of  scale,  in  the  near 
future.  When  afterward  I  found  myself  able 
to  resume  the  study,  there  seemed  no  im- 
mediate prospect  of  resuming  experimentation, 
and  it  was  thought  best  to  give  the  material 
comparatively  full  treatment.  It  will  readily 
be  understood  from  this  account  that  I  am 
responsible  for  the  planning  of  the  experimental 


work  as  well  as  for  the  discussion  of  results 
here  contained,  while  Mr.  Murphy  is  responsible 
for  the  experimental  work.  It  must  not  be 
understood,  however,  that  in  assuming  responsi- 
bility for  the  discussion  I  also  claim  sole  credit 
for  what  is  novel  in  the  generalizations.  Many 
conclusions  were  reached  by  us  jointly  during 
our  association,  and  others  were  developed  by 
Mr.  Murphy  in  his  report.  These  have  been 
incorporated  in  the  present  report,  so  far  as 
they  appeared  to  be  sustained  by  the  more 
elaborate  analysis,  and  specific  credit  is  given 
only  where  I  found  it  practicable  to  quote 
from  Mr.  Murphy's  manuscript. 

Mr.  J.  A.  Burgess  was  for  a  short  time  a 
scientific  assistant  in  the  laboratory,  and  his 
work  is  described  in  another  connection. 
Credit  should  be  given  to  Mr.  L.  E.  Eshleman, 
carpenter,  and  Mr.  Waldemar  Arntzen,  mechan- 
ician, for  excellent  work  in  the  construction  of 
apparatus.  I  recall  with  sincere  gratitude 
the  cordial  cooperation  of  several  members  of 
the  university  faculty,  and  the  investigation  is 
especially  indebted  to  the  good  offices  and 
technical  knowledge  of  Prof.  S.  B.  Christy 
and  Prof.  J.  N.  Le  Conte. 

Portions  of  my  manuscript  were  read  by 
Dr.  R.  S.  Woodward  and  Dr.  Lyman  J.  Briggs, 
and  the  entire  manuscript  was  read  by  Mr.  C.  E. 
Van  Orstrand  and  Mr.  Willard  D.  Johnson. 
To  these  gentlemen  and  to  members  of  the 
editorial  staff  of  the  Geological  Survey  I  am 
indebted  for  criticisms  and  suggestions  leading 
to  the  elimination  of  some  of  the  crudities  of 
the  original  draft. 

While  the  aid  which  my  work  has  received 
from  many  colleagues  has  been  so  kindly  and 
efficient  that  individual  mention  seems  invid- 
ious, my  gratitude  must  nevertheless  be 
expressed  for  valuable  assistance  by  Mr.  Fran- 
cois E.  Matthes  in  the  examination  of  foreign 
literature,  and  for  the  unfailing  encouragement 
and  support  of  Mr.  M.  O.  Leighton,  until 
recently  in  charge  of  the  hydrographic  work 
of  the  Survey. 

G.  K.  G. 


ABSTRACT. 


Scope. — The  finer  debris  transported  by  a 
stream  is  borne  in  suspension.  The  coarser  is 
swept  along  the  channel  bed.  The  suspended 
load  is  readily  sampled  and  estimated,  and 
much  is  known  as  to  its  quantity.  The  bed 
load  is  inaccessible  and  we  are  without  definite 
information  as  to  its  amount.  The  primary 
purpose  of  the  investigation  was  to  learn  the 
laws  which  control  the  movement  of  bed  load, 
and  especially  to  determine  how  the  quantity 
of  load  is  related  to  the  stream's  slope  and  dis- 
charge and  to  the  degree  of  comminution  of  the 
debris. 

Method. — To  this  end  a  laboratory  was 
equipped  at  Berkeley,  Cal.,  and  experiments 
were  performed  in  which  each  of  the  three  con- 
ditions mentioned  was  separately  varied  and 
the  resulting  variations  of  load  were  observed 
and  measured.  Sand  and  gravel  were  sorted 
by  sieves  into  grades  of  uniform  size.  Deter- 
minate discharges  were  used.  In  each  experi- 
ment a  specific  load  was  fed  to  a  stream  of 
specific  width  and  discharge,  and  measurement 
was  made  of  the  slope  to  which  the  stream 
automatically  adjusted  its  bed  so  as  to  enable 
the  current  to  transport  the  load. 

The  slope  factor. — For  each  combination  of 
discharge,  width,  and  grade  of  de'bris  there  is  a 
slope,  called  competent  slope,  which  limits 
transportation.  With  lower  slopes  there  is  no 
load,  or  the  stream  has  no  capacity  J  for  load. 
With  higher  slopes  capacity  exists;  and 
increase  of  slope  gives  increase  of  capacity. 
The  value  of  capacity  is  approximately  propor- 
tional to  a  power  of  the  excess  of  slope  above 
competent  slope.  If  S  equal  the  stream's  slope 
and  a  equal  competent  slope,  then  the  stream's 
capacity  varies  as  (S  —  a)n.  This  is  not  a  de- 
ductive, but  an  empiric  law.  The  exponent  n 
has  not  a  fixed  value,  but  an  indefinite  series 
of  values  depending  on  conditions.  Its  range 
of  values  in  the  experience  of  the  laboratory 

1  Capacity  is  defined  for  the  purposes  of  this  paper  as  the  maximum 
load  of  a  given  kind  of  de'bris  which  a  given  stream  can  transport.  See 
page  35. 

10 


is  from  0.93  to  2.37,  the  values  being  greater 
as  the  discharges  are  smaller  or  the  de'bris  is 
coarser. 

The  discharge  factor. — For  each  combination 
of  width,  slope,  and  grade  of  de'bris  there  is  a 
competent  discharge,  «.  Calling  the  stream's 
discharge  Q,  the  stream's  capacity  varies  as 
(Q  —  K)°.  The  observed  range  of  values  for  o 
is  from  0.81  to  1.24,  the  values  being  greater 
as  the  slopes  are  smaller  or  the  debris  is 
coarser.  Under  like  conditions  o  is  less  than 
n;  or,  in  other  words,  capacity  is  less  sensitive 
to  change3  of  discharge  than  to  changes  of 
slope. 

The  fineness  factor. — For  each  combination 
of  width,  slope,  and  discharge  there  is  a  limit- 
ing fineness  of  de'bris  below  which  no  transpor- 
tation takes  place.  Calling  fineness  (or  degree 
of  comminution)  F  and  competent  fineness  <j>, 
the  stream's  capacity  varies  with  (F  — 0)P. 
The  observed  range  of  values  for  p  is  from  0.50 
to  0.62,  the  values  being  greater  as  slopes  and 
discharges  are  smaller.  Capacity  is  less  sensi- 
tive to  changes  in  fineness  of  de'bris  than  to 
changes  in  discharge  or  slope. 

The  form  factor. — Most  of  the  experiments 
were  with  straight  channels.  A  few  with 
crooked  channels  yielded  nearly  the  same  esti- 
mates of  capacity.  The  ratio  of  depth  to  width 
is  a  more  important  factor.  For  any  combi- 
nation of  slope,  discharge,  and  fineness  it  is 
possible  to  reduce  capacity  to  zero  by  making 
the  stream  very  wide  and  shallow  or  very  nar- 
row and  deep.  Between  these  extremes  is  a 
particular  ratio  of  depth  to  width,  p,  corre- 
sponding to  a  maximum  capacity.  The  values 
of  p  range,  under  laboratory  conditions,  from 
0.5  to  0.04,  being  greater  as  slope,  discharge, 
and  fineness  are  less. 

Velocity. — The  velocity  which  determines 
capacity  for  bed  load  is  that  near  the  stream's 
bed,  but  attempts  to  measure  bed  velocity 
were  not  successful.  Mean  velocity  was  meas- 
ured instead.  To  make  a  definite  comparison 
between  capacity  and  mean  velocity  it  is  neces- 


ABSTRACT. 


11 


sary  to  postulate  constancy  in  some  accessory 
condition.  If  slope  be  the  constant,  in  which 
case  velocity  changes  with  discharge,  capacity 
varies  on  the  average  with  the  3.2  power  of 
velocity.  If  discharge  be  the  constant,  in 
which  case  velocity  changes  with  slope,  capacity 
varies  on  the  average  with  the  4.0  power  of 
velocity.  If  depth  be  the  constant,  in  which 
case  velocity  changes  with  simultaneous 
changes  of  slope  and  discharge,  capacity  varies 
on  the  average  with  the  3.7  power  of  velocity. 
The  power  expressing  the  sensitiveness  of 
capacity  to  changes  of  mean  velocity  has  in 
each  case  a  wide  range  of  values,  being  greater 
as  slope,  discharge,  and  fineness  are  less. 

Mixtures. — In  general,  debris  composed  of 
particles  of  a  single  size  is  moved  less  freely 
than  debris  containing  particles  of  many  sizes. 
If  fine  material  be  added  to  coarse,  not  only  is 
the  total  load  increased  but  a  greater  quantity 
of  the  coarse  material  is  carried. 

Modes  of  transportation;  movement  of  par- 
ticles.— Some  particles  of  the  bed  load  slide; 
many  roll;  the  multitude  make  short  skips  or 
leaps,  the  process  being  called  saltation.  Sal- 
tation grades  into  suspension.  When  particles 
of  many  sizes  are  moved  together  the  larger 
ones  are  rolled. 

Modes  of  transportation;  collective  move- 
ment.— When  the  conditions  are  such  that  the 
bed  load  is  small,  the  bed  is  molded  into  hills, 
called  dunes,  which  travel  downstream.  Their 
mode  of  advance  is  like  that  of  eolian  dunes, 
the  current  eroding  their  upstream  faces  and 
depositing  the  eroded  material  on  the  down- 
stream faces.  With  any  progressive  change  of 
conditions  tending  to  increase  the  load,  the 
dunes  eventually  disappear  and  the  de'bris  sur- 
face becomes  smooth.  The  smooth  phase  is  in 
turn  succeeded  by  a  second  rhythmic  phase,  in 
which  a  system  of  hills  travel  upstream.  These 
are  called  antidunes,  and  their  movement  is 
accomplished  by  erosion  on  the  downstream 
face  and  deposition  on  the  upstream  face. 
Both  rhythms  of  de'bris  movement  are  initiated 
by  rhythms  of  water  movement. 

Application  of  formulas. — While  the  prin- 
ciples discovered  in  the  laboratory  are  neces- 
sarily involved  in  the  work  of  rivers,  the  labo- 
ratory formulas  are  not  immediately  available 
for  the  discussion  of  river  problems.  Being 
both  empiric  and  complex,  they  will  not  bear 


extensive  extrapolation.  Under  some  circum- 
stances they  may  be  used  to  compare  the  work 
of  one  stream  with  that  of  another  stream  of 
the  same  type,  but  they  do  not  permit  an  esti- 
mate of  a  river's  capacity  to  be  based  on  the 
determined  capacities  of  laboratory  streams. 
The  investigation  made  an  advance  in  the 
direction  of  its  primary  goal,  but  the  goal  was 
not  reached. 

Load  versus  energy. — The  energy  of  a  stream 
is  measured  by  the  product  of  its  discharge 
(mass  per  unit  tune),  its  slope,  and  the  accel- 
eration of  gravity.  In  a  stream  without  load 
the  energy  is  expended  in  flow  resistances, 
which  are  greater  as  velocity  and  viscosity  are 
greater.  Load,  including  that  carried  in  sus- 
pension and  that  dragged  along  the  bed,  affects 
the  energy  in  three  ways.  (1)  It  adds  its  mass 
to  the  mass  of  the  water  and  increases  the 
stock  of  energy  pro  rata.  (2)  Its  transporta- 
tion involves  mechanical  work,  and  that  work 
is  at  the  expense  of  the  stream's  energy.  (3) 
Its  presence  restricts  the  mobility  of  the  water, 
in  effect  increasing  its  viscosity,  and  thus  con- 
sumes energy.  For  the  finest  elements  of  load 
the  third  factor  is  more  important  than  the 
second;  for  coarser  elements  the  second  is  the 
more  important.  For  each  element  the  second 
and  third  together  exceed  the  first,  so  that  the 
net  result  is  a  tax  on  the  stream's  energy. 
Each  element  of  load,  by  drawing  on  the  supply 
of  energy,  reduces  velocity  and  thus  reduces 
capacity  for  all  parts  of  the  load.  This  prin- 
ciple affords  a  condition  by  which  total  capacity 
is  limited.  Subject  to  this  condition  a  stream's 
load  at  any  time  is  determined  by  the  supply 
of  de'bris  and  the  fineness  of  the  available 
kinds. 

Flume  transportation. — In  the  experiments 
described  above — experiments  illustrating 
stream  transportation — the  load  traversed  a 
plastic  bed  composed  of  its  own  material. 
Other  experiments  were  arranged  in  which  the 
load  traversed  a  rigid  bed,  the  bottom  of  a 
flume.  Capacities  are  notably  larger  for  flume 
transportation  than  for  stream  transportation, 
and  their  laws  of  variation  are  different. 
Rolling  is  an  important  mode  of  progression. 
For  rolled  particles  the  capacity  increases  with 
coarseness,  for  leaping  particles  with  fineness. 
Capacity  increases  with  slope  and  usually  with 
discharge  also,  but  the  rates  of  increase  are  less 


12 


TRANSPORTATION    OF   DEBRIS  BY   RUNNING   WATER. 


than    in   stream   transportation.     Capacity  is 
reduced  by  roughness  of  bed. 

Vertical  velocity  curve. — The  vertical  distri- 
bution of  velocities  in  a  current  is  controlled 
by  conditions.  The  level  of  maximum  velocity 
may  have  any  position  in  the  upper  three- 
fourths  of  the  current.  In  loaded  streams  its 
position  is  higher  as  the  load  is  greater.  In 
unloaded  streams  its  position  is  higher  as  the 


slope  is  steeper,   as  the  discharge  is  greater, 
and  as  the  bed  is  rougher. 

Pilot  tube. — The  constant  of  the  Pitot  veloc- 
ity gage — the  ratio  between  the  head  realized 
and  the  theoretic  velocity  head — is  not  the 
same  in  all  parts  of  a  conduit,  being  less  near 
the  water  surface  and  greater  near  the  bottom 
or  side  of  the  conduit. 


NOTATION. 


Certain  letters  are  used  continuously  in  the 
volume  as  symbols  for  quantities,  a  definition 
accompanying  the  first  use.  These  are  ar- 
ranged alphabetically  in  the  following  list, 
with  brief  characterization  and  page  reference. 
The  list  does  not  include  letters  used  tem- 
porarily as  symbols  and  defined  in  immediate 
connection  with  their  use;  and  if  the  same 
letter  has  both  temporary  and  continuous  uses 
only  the  continuous  use  is  here  given. 

Page. 

A A  constant  of  the  Pitot-Darcy  gage.       254 

(A),  (B),  etc Grades    of   debris,    separated    by 

sieves 21 

(A,  G4),  etc Mixtures  of  debris  grades,  the  let- 
ters and  figures  indicating  com- 
ponents and  proportions 169 

a A  pressure  constant  (Lechalas) ...       193 

<T Numerical    constant  in  equation 

C=b2(\-ctR)Rm 125 

Accents  (' x  A  v ) .  An  accent  over  the  symbol  of  a 
variable  indicates  the  nature  of 
its  influence  on  a  function. 
Thus  A=f(6,  £,  t),  E)  states 
that  A  is  an  increasing  function 
of  B,  a  decreasing  function  of  C, 
a  maximum  function  of  D,  and  a 

minimum  function  of  -B 96 

6 A  constant  in  equation  (109) 191 

6' A  constant  depth  in  equation  (21) .        88 

£>i A  constant  capacity  in  equation 

(10) 64 

&2 A  constant  capacity  in  equation 

(54) 129 

63 A  constant  in  equation  (64) 139 

*4 A  constant  in  equation  (75) 151 

&5 A  constant  in  equation  (91) 186 

C Capacity  of  a  stream  for  traction  of 

debris  (in  gm. /sec.) 35 

CT Readjusted  value  of  capacity. . .  141, 151 

c Constant  coefficient  in  y=cx1 99 

c, Constant  coefficient  in  C=e1Sil ...      109 

D Mean  diameter  (in  feet)  of  particles 

of  debris 21 

d Depth  of  current  (in  feet) 33 

d Differential  97 

E Efficiency  of  stream  for  traction  of 

debris;  capacity  per  unit  dis- 
charge per  unit  slope;  CjQS 36 

e Base  of  Naperian  logarithms 61 


F Linear  fineness  of  debris;  the  re- 
ciprocal of  /) 21, 183 

FI Bulk  fineness  of  debris;  the  num- 
ber of  particles  in  a  cubic  foot.  21, 183 

ft. /sec Feet  per  second;  unit  of  velocity..        34 

f t.3/sec Cubic  feet  per  second ;  unit  of  dis- 
charge         34 

g Acceleration  of  gravity 225 

gm./sec Grams  per  second ;  unit  of  load  and 

of  capacity  for  load 34 

B Range  of  fineness 96 

B,  HI Readings    of    comparator,    Pitot- 

Darcygage 254 

I Synthetic  index  of  relative  varia- 
tion          99 

I\ /for  capacity  and  slope 122 

la /for  capacity  and  discharge 147 

/i /for  capacity  and  fineness 153 

fas /for  capacity  and    depth,  slope 

being  constant 164 

/rf« /for  capacity  and  depth,  discharge 

being  constant 164 

far /for    capacity  and   depth,   mean 

velocity  being  constant 164 

/< /for  efficiency  and  slope 122 

Is /for  capacity  and  slope,  form  ratio 

being  constant 119 

Iv /for  capacity  and  mean  velocity.       157 

Ivs /for  capacity  and  mean  velocity, 

slope  being  constant 157 

/K« /for  capacity  and  mean  velocity, 

discharge  being  constant 157 

Ivd /for  capacity  and  mean  velocity, 

depth  being  constant 157 

Iw /  for  capacity  and  slope,  width 

being  constant 119 

i Index   of   relative    variation;  ex- 
ponent in  y=vx* 99 

t'i i  for  capacity  and  slope 99 

j'2 ifor  capacity  and  form  ratio 130 

i3 ifor  capacity  and  discharge 141 

t4 i  for  capacity  and  fineness 153 

j Exponent  in^=er' 99 

;, Exponent  in  C=c1Sh 109 

K,  I- Constants  of  the  Pitot-Darcy  gage. .      254 

K A  constant  discharge,  correspond- 
ing to  competent  discharge 139 

L Length,  as  a  dimension  of  units. .       139 

L Load;  mass  of  debris  transported 

through  a  cross  section  per  unit 

time  (in  gm./sec.) 35 

M. Mass,  as  a  dimension  of  units.  . . .       139 

13 


14 


TRANSPORTATION    OF   DEBRIS   BY   EUNNING   WATER. 


Page. 

m  ...............  Exponent  in  0=63(1  -<rR)  .ft1*....      129 

m,  ...............  Exponent;  =m+l  .............  128,129 


n  ...............  Exponent  in  C=bt(S-o)  "....  61,64,96 

n,  ...............  Exponent  in  d=—^ 


88 


o  ...............  Exponent  in  C=ba(Q-x)  °  .......  139 

p  ...............  Exponent  in  C=bt(F-<j>)p  ......  151 

p.  e  ..............  Probable  error  ..............  .  .....  89 

T.  ...............  Ratio  of  circumference  of  circle  to 

diameter  .....................  21 

<j>  ...............  A  constant  linear  fineness,  corre- 

sponding to  competent  fineness.  151 

Q  ...............  Discharge  (in  ft.3/sec.)  ............  35 

R  ................  Form  ratio;  djw  ..................  36 

p  ...............  Value  of  R  corresponding  to  maxi- 

mum capacity  ..................  129 

5  ................  Slope,  in  per  cent,  of  stream  bed 

or  water  surface  ................  34 

a  ......  .  ..........  Slope  of  stream  bed  or  water 

surface;  =  fall  per  unit  distance  .  34 


Page. 
a A  constant  slope,  corresponding  to 

competent  slope 64 

£ Sum  of 228 

T Time,  aa  a  dimension  of  units. . .  139 

U. Duty  of  water  (in  gm./sec.)  for 

traction  of  debris;  capacity  per 

unit  discharge;  C/Q 36 

V. Velocity  of  current  (in  ft. /sec.). . . .  163 

T'6 Velocity  of  stream  at  contact  with 

bed  (Lechalas) 193 

Ym Mean  velocity;  discharge-Harea  of 

cross  section;  Q/dw 33,  94, 155 

| ', Velocity  of  stream  at  surface 

(Lechalas) 194 

i> Variable  coefficient  in  y=vxi 99 

r, Variable  coefficient  in  C=v1S'1  ...  99 

r3 Variable  coefficient  in  C=v3Q  3  .  . .  141 

(/• Width  of  stream  channel  (in  feet) . .  67 


THE  TRANSPORTATION  OF  DEBRIS  BY  RUNNING  WATER. 


By  GROVE  KARL  GILBERT. 


CHAPTER  I.— THE  OBSERVATIONS. 


INTRODUCTION. 

GENERAL    CLASSIFICATION. 

Streams  of  water  carry  forward  debris  in 
various  ways.  The  simplest  is  that  in  which 
the  particles  are  slidden  or  rolled.  Sliding 
rarely  takes  place  except  where  the  bed  of  the 
channel  is  smooth.  Pure  rolling,  in  which  the 
particle  is  continuously  in  contact  with  the 
bed,  is  also  of  small  relative  importance.  If 
the  bed  is  uneven,  the  particle  usually  does  not 
retain  continuous  contact  but  makes  leaps,  and 
the  process  is  then  called  saltation,  an  expres- 
sive name  introduced  by  McGee.1  With  swifter 
current  leaps  are  extended,  and  if  a  particle 
thus  freed  from  the  bed  be  caught  by  an 
ascending  portion  of  a  swirling  current  its 
excursion  may  be  indefinitely  prolonged.  Thus 
borne  it  is  said  to  be  suspended,  and  the  process 
by  which  it  is  transported  is  called  suspension. 
There  is  no  sharp  line  between  saltation  and 
suspension,  but  the  distinction  is  nevertheless 
important,  for  it  serves  to  delimit  two  methods 
of  hydraulic  transportation  which  follow  differ- 
ent laws.  In  suspension  the  efficient  factor  is 
the  upward  component  of  motion  in  parts  of  a 
complex  current.  In  other  transportation, 
including  saltation,  rolling,  and  sliding,  the 
efficient  factor  is  the  motion  parallel  with  the 
bed  and  close  to  it.  This  second  division  of 
current  transportation  is  called  by  certain 
French  engineers  entrainement  but  has  received 
no  name  in  English.  Being  in  need  of  a  suc- 
cinct title,  I  translate  the  French  designation, 
which  indicates  a  sweeping  or  dragging  along, 
by  the  word  traction,  thus  classifying  hydraulic 
transportation  as  (1)  hydraulic  suspension  and 
(2)  hydraulic  traction. 

i  McGee,  W  J,  Oeol.  Soc.  America  Bull.,  vol.  19,  p.  199, 1908. 


The  bed  of  a  natural  stream  which  carries  a 
large  load  of  debris  is  composed  of  loose  grains 
identical  in  character  with  those  transported. 
The  material  of  the  load  is  derived  from  and 
returned  to  the  bed,  and  the  surface  of  the  bed 
is  molded  by  the  current.  When  debris  is 
transported  through  artificial  channels,  such  as 
flumes  and  pipes,  the  bed  is  usually  rigid  and 
unyielding.  Trifling  as  this  difference  appears, 
it  yet  occasions  a  marked  contrast  in  the  quan- 
titative laws  of  transportation,  and  in  the  labo- 
ratory the  two  kinds  of  transportation  were  the 
subjects  of  separate  courses  of  experimentation. 
It  is  necessary,  therefore,  for  present  purposes, 
to  base  a  second  classification  of  hydraulic 
transportation  on  the  nature  of  the  bottom. 
As  the  bed  is  typically  plastic  in  stream  chan- 
nels and  typically  rigid  in  flumes  and  other 
artificial  channels,  it  is  convenient  to  call  the 
two  classes  stream  transportation  and  flume 
transportation . 

The  second  classification  traverses  the  first 
and  their  combination  gives  four  divisions — 
stream  suspension,  stream  traction,  flume  sus- 
pension, and  flume  traction.  This  report 
treats  of  stream  traction  and  flume  traction. 
It  contains  the  record  .  and  discussion  of  a 
series  of  experiments  made  hi  a  specially 
equipped  laboratory  at  the  University  of  Cali- 
fornia, Berkeley,  in  the  years  1907-1909. 

STREAM    TRACTION. 

Previous  to  the  Berkeley  work  little  was 
known  of  the  quantitative  laws  of  stream 
traction.  The  quantity  of  material  trans- 
ported has  sometimes  been  said  to  be  propor- 
tional to  the  square  of  the  slope,  but  I  have 
failed  to  discover  that  the  statement  has  a  re- 
corded basis  in  theory  or  observation.  A  state- 

15 


16 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


ment  more  frequently  encountered  is  to  the 
effect  that  the  quantity  varies  with  the  sixth 
power  of  the  velocity;  and  the  origin  of  this 
assertion  is  not  in  doubt.  It  is  an  erroneous 
version  of  a  deductive  law  commonly  attribu- 
ted to  Hopkins  (1844)  or  Airy  and  Law  (1885), 
although  announced  as  early  as  1823  by  Les- 
lie.1 The  law,  as  formulated  by  Hopkins,  is 
that  "  the  moving  force  of  a  current,  estimated 
by  the  volume  or  weight  of  the  masses  of  any 
proposed  form  which  it  is  capable  of  moving, 
varies  as  the  sixth  power  of  the  velocity"; 
and  this  law  pertains  not  at  all  to  the  quantity 
of  material  moved,  but  to  the  maximum  size 
of  the  grain  or  pebble  or  bowlder  a  given  cur- 
rent is  competent  to  move. 

The  subject  of  the  competence  of  currents, 
or  the  relation  of  velocities  to  the  size  of  par- 
ticles they  can  move,  has  also  been  treated 
experimentally  by  several  investigators,  and 
some  account  of  their  work  will  be  given  in 
later  chapters. 

About  the  year  1883  Deacon  made,  in  Man- 
chester, England,  a  notable  series  of  experi- 
ments in  the  field  of  stream  traction.  As  a 
result  of  definite  measurements  of  quantities 
of  sand  transported  and  of  the  velocities  of  the 
transporting  currents,  he  announced  2  that  the 
amount  transported,  instead  of  varying  with 
the  sixth  power  of  the  velocity,  as  had  been 
supposed,  actually  varies  with  the  fifth  power. 

In  the  field  of  flume  traction  the  work  has 
been  somewhat  more  extensive,  having  as  its 
special  incentive  the  needs  arising  in  ore  mills 
for  the  conveyance  of  crushed  rock;  and  a 
resumS  of  results  will  be  found  in  the  chapter 
on  flume  traction. 

A  still  greater  body  of  investigation  has  been 
conducted  by  German  and  French  engineers 
with  the  use  of  laboratory  models  of  river  chan- 
nels. The  French  work  was  done  largely  for 
the  purpose  of  testing  certain  rules  formulated 
by  Fargue 3  for  the  improvement  of  navigable 
streams.  The  German  experiments  were  and 
still  are  addressed  to  the  broader  subject  of 

i  Sir  John  Leslie's  analysis  is  to  be  found  in  his  Elements  of  natural 
philosophy,  and  in  the  edition  of  1829  occurs  at  pages  426-427  of  volume  1. 
David  Stevenson  mentions  1823,  which  probably  indicates  the  first 
edition.  William  Hopkins  gives  a  diflerent  analysis  with  practically  the 
same  result,  and  does  not  mention  Leslie.  The  passage  quoted  occurs  in 
Cambridge  Philcs.  Sec.  Trans.,  vol.  8,  p.  233, 1844,  and  is  probably  from 
his  earliest  discussion  of  the  subject.  Wilfred  Airy's  later  but  evidently 
independent  analysis  appears  in  Inst.  Civil  Eng.  Proc.,  vol.  82,  pp.  25-26, 
1885,  with  expansion  by  Henry  Law  on  pp.  29-30. 

»  Deacon,  G.  F.,  Inst.  Civil  Eng.  Proc.,  vol.  98,  pp.  93-%,  1894. 

*  Fargue,  L.,  Annales  des  ponts  et  chauss&s,  1894. 


river  engineering  in  general  and  include  within 
their  scope  the  scientific  study  of  the  ways  in 
which  rivers  shape  and  reshape  their  channels. 
The  quantitative  laws  of  stream  traction, 
which  constitute  the  chief  theme  of  the  Berke- 
ley work,  thus  fall  within  the  province  of  the 
German  investigators,  but  their  study  has  not 
been  taken  up.  There  are  three  German  labo- 
ratories, all  well  equipped,  located  severally 
at  Dresden,  Karlsruhe,  and  Berlin.4 

The  flow  of  a  stream  is  a  complex  process, 
involving  interactions  which  have  thus  far 
baffled  mechanical  analysis.  Stream  traction 
is  not  only  a  function  of  stream  flow  but  itself 
adds  a  complication.  Some  realization  of  the 
complexity  may  be  achieved  by  considering 
briefly  certain  of  the  conditions  which  modify 
the  capacity  of  a  stream  to  transport  debris 
along  its  bed.  Width  is  a  factor;  a  broad 
channel  carries  more  than  a  narrow  one. 
Velocity  is  a  factor;  the  quantity  of  debris 
carried  varies  greatly  for  small  changes  in  the 
velocity  along  the  bed.  Bed  velocity  is  affected 
by  slope  and  also  by  depth,  increasing  with  each 
factor;  and  depth  is  affected  by  discharge  and 
also  by  slope.  If  there  is  diversity  of  velocity 
from  place  to  place  over  the  bed,  more  debris  is 
carried  than  if  the  average  velocity  everywhere 
prevails,  and  the  greater  the  diversity  the 
greater  the  carrying  power  of  the  stream.  Size 
of  transported  particles  is  a  factor,  a  greater 
weight  of  fine  debris  being  carried  than  of 
coarse.  The  density  of  debris  is  a  factor,  a 
low  specific  gravity  being  favorable.  The 
shapes  of  particles  affect  traction,  but  the 
nature  of  this  influence  is  not  well  understood. 
An  important  factor  is  found  in  form  of  chan- 
nel, efficiency  being  affected  by  turns  and  curv- 
ature and  also  by  the  relation  of  depth  to 
width.  The  friction  between  current  and 
banks  is  a  factor  and  therefore  likewise  the 
nature  of  the  banks.  So,  too,  is  the  viscosity 
of  the  water,  a  property  varying  with  tempera- 
ture and  also  with  impurities,  whether  dis- 
solved or  suspended. 

The  enumeration  might  be  extended,  com- 
plexity might  be  further  illustrated  by  pointing 

<  The  equipment  and  work  of  the  laboratory  of  river  engineering  of  the 
Technical  High  School  of  Dresden  are  discussed  in  the  Zeitschrift  fur 
Bauwesen,  vol.  50,  pp.  343-300,  1900,  and  vol.  55,  pp.  664-C80,  1905;  the 
equipment  of  the  laboratory  of  the  Technical  High  School  "Frederici- 
ana  "  of  Karlsruhe  in  the  same  journal,  vol.  53,  pp.  103-136,  1903,  and  vol. 
60,  pp.  313-328, 1910;  and  the  equipment  and  work  of  the  Laboratory  for 
River  Improvement  and  Naval  Architecture,  Berlin,  in  vol.  56,  pp.  123- 
151,  323-324,  1906. 


THE   OBSERVATIONS. 


17 


out  the  influence  of  conditions  on  one  another, 
and  the  difficulty  of  measuring  the  detrital 
loads  of  streams  might  be  dwelt  upon,  but 
enough  has  been  said  to  warrant  the  statement 
that  an  adequate  analysis  with  quantitative 
relations  can  not  be  achieved  by  the  mere  obser- 
vation of  streams  in  their  natural  condition. 
It  is  necessary  to  supplement  such  observation 
by  experiments  in  which  the  conditions  are 
definitely  controlled. 

OUTLINE     OF     COURSE      OF     EXPERIMENTATION. 

In  the  work  of  the  Berkeley  laboratory 
capacity  for  hydraulic  traction  was  compared 
with  discharge,  with  slope,  depth,  and  width 
of  current,  and  with  fineness  of  debris;  and 
minor  attention  was  given  to  velocity  and  to 
curvature  of  channel.  For  the  principal  ex- 
periments a  straight  trough  was  used,  the  sides 
being  vertical  and  parallel,  the  ends  open,  the 
bottom  plane  and  horizontal.  Through  this  a 
stream  of  water  was  run,  the  discharge  being 
controlled  and  measured.  Xear  the  head  of 
the  trough  sand  was  dropped  into  the  water  at 
a  uniform  rate,  the  sand  grains  being  of  approxi- 
mately uniform  size.  At  the  beginning  of  an 
experiment  the  sand  accumulated  in  the  trough, 
being  shaped  by  the  current  into  a  deposit  with 
a  gentle  forward  slope.  The  deposit  gradually 
extended  to  the  outfall  end  of  the  trough,  and 
eventually  accumulation  ceased,  the  rate  at 
•which  sand  escaped  at  the  outfall  having  be- 
come equal  to  the  rate  at  which  it  was  fed 
above.  The  slope  was  thus  automatically 
adjusted  and  became  just  sufficient  to  enable 
the  particular  discharge  to  transport  the  par- 
ticular quantity  of  the  particular  kind  of  sand. 
The  slope  was  then  measured.  Measurement 
was  made  also  of  the  depth  of  the  current ;  and 
the  mean  velocity  was  computed  from  the  dis- 
charge, width,  and  depth. 

In  a  second  experiment,  with  the  same  dis- 
charge, the  sand  was  fed  to  the  current  at  a 
different  rate,  and  the  resulting  slope  and 
depth  were  different.  By  a  series  of  such  ex- 
periments was  developed  a  law  of  relation 
between  the  quantity  of  sand  carried,  or  the 
load,  and  the  slope  necessary  to  carry  it,  this 
law  pertaining  to  the  particular  discharge  and 
the  particular  grade  of  sand.  The  same  ex- 
periments showed  also  the  relations  of  the 
velocity  of  the  current  to  slope  and  load. 
20921°— Xo.  80 — 14 2 


Another  series  of  experiments,  employing  a 
greater  or  a  less  discharge,  gave  a  parallel  set 
of  relations  between  slope,  load,  and  velocity. 
By  multiplying  such  series  the  relations  be- 
tween discharge  and  slope,  discharge  and 
load,  and  discharge  and  velocity  were  de- 
veloped. 

Then  a  third  condition  was  varied,  the 
width  of  channel;  and  finally  the  remaining 
condition  under  control,  the  size  of  the  sand 
grains.  Thus  data  were  obtained  for  studying 
the  quantitative  relations  between  load,  slope, 
discharge,  width,  and  fineness,  as  well  as  the 
relations  of  depth  and  mean  velocity  to  all 
others.  In  all,  the  range  of  conditions  in- 
cluded six  discharges,  six  widths  of  channel, 
and  eight  grades  of  sand  and  gravel,  but  not 
all  the  possible  combinations  of  these  were 
made.  The  actual  number  of  combinations 
was  130,  and  under  each  of  these  were  a  series 
of  measurements  of  load,  slope,  and  depth. 
There  were  also  limited  series  of  experiments 
involving  a  greater  number  of  discharges  and 
a  greater  number  of  widths.  The  separate 
determinations  of  load  and  slope  numbered 
nearly  1,200,  and  those  of  depth  about  900. 

SCOPE    OF   EXPERIMENTS. 

Before  proceeding  to  a  fuller  description  of 
apparatus  and  experiments,  let  us  consider  to 
what  extent  the  conditions  of  the  laboratory 
were  representative  of  the  conditions  which 
exist  in  the  natural  stream. 

The  sand  used  came  from  the  beds  of  Ameri- 
can and  Sacramento  rivers  and  was  assumed  to 
be  representative  of  river  sand  in  general. 
No  attention  was  paid  to  the  influence  on 
traction  of  the  form  and  density  of  grains. 
Each  sample  used  was  separated  from  the 
natural  mixture  by  means  of  two  sieves  and 
was  composed  of  grains  which  passed  through 
a  certain  mesh  and  were  arrested  by  a  mesh 
slightly  smaller.  In  the  sand  carried  by  a 
river  near  its  bed  the  range  of  size  is  much 
wider.  The  limit  of  coarseness  is  found  in 
those  particles  which  the  current  is  barely 
able  to  roll,  the  limit  in  fineness  in  those  par- 
ticles which  the  swirls  of  the  current  are  not 
quite  able  to  lift  into  suspension;  and  the 
limits  vary  from  point  to  point  of  the  channel 
bed.  This  difference  in  condition  was  not 
wholly  ignored,  but  a  short  supplementary 


18 


TRANSPORTATION    OF    DEBRIS   BY    RUNNING    WATER. 


series  of  experiments  was  made  with  definite 
mixtures  of  sand  of  various  sizes,  as  well  as 
with  a  natural  mixture. 

The  straight  channel  of  the  laboratory  differs 
materially  from  the  curved  channels  of  nature. 
It  gives  comparatively  uniform  depths  and 
velocities  from  side  to  side  and  from  point  to 
point  in  the  direction  of  the  flow,  while  in  a 
curved  channel  the  depths  and  velocities  vary 
greatly  both  across  and  along  the  channel. 
This  difference  in  condition  also  received  some 
attention,  a  short  series  of  experiments  being 
made  with  crooked  and  curved  channels. 

The  vertical  sides  of  the  troughs  did  not  well 
represent  the  sloping  banks  of  rivers,  and  no 
attempt  was  made  to  measure  the  qualification 
due  to  this  difference.  The  cross  section  of 
the  laboratory  current  was  essentially  a  rec- 
tangle and  the  ability  of  the  current  to  trans- 
port was  found  to  be  definitely  related  to  the 
ratio  between  depth  and  width;  but  satisfactory 
connection  was  not  made  between  this  relation 
and  the  forms  of  cross  section  in  rivers. 

The  thalweg  of  a  river  channel  traverses  an 
alternation  of  deeps  and  shoals,  the  deeps  being 
characterized  by  a  different  system  of  velocities 
and  by  a  different  line  of  separation  between 
the  grades  of  debris  carried  severally  by  sus- 
pension and  traction;  but  these  contrasts  were 
touched  only  in  a  qualitative  way  in  the  work 
of  the  laboratory. 

Each  experiment  dealt  with  a  slope  in  ad- 
justment with  a  particular  discharge  and  a  par- 
ticular grade  of  sand.  In  a  natural  stream  the 
discharge  is  subject  to  variation,  and  its 
changes  cause  changes  in  the  fineness  of  the 
material  carried  along  the  bottom.  Load  and 
the  local  slopes  are  ever  in  process  of  adjust- 
ment to  the  temporary  conditions  of  discharge 
and  fineness,  but  the  adjustment  is  never  com- 
plete. The  general  or  average  slope  is  adjusted 
to  an  indeterminate  discharge  which  is  neither 
the  smallest  nor  the  greatest.  For  this  phase 
of  disparity  allowance  is  not  easily  made. 

One  of  the  conditions  affecting  velocities  is 
friction  on  bed  and  sides  of  channel.  Friction 
on  the  bed  depends  partly  on  the  roughness  of 
the  bed  and  partly  on  the  consumption  of 
energy  by  traction.  Its  laws  are  the  same  in 
laboratory  and  in  river.  Friction  on  the  sides 
depends  on  the  character  of  the  channel  wall 
and  must  be  materially  greater  on  river  banks 
than  on  the  smooth  sides  of  the  experiment 


trough.  The  magnitude  of  the  difference  was 
not  determined.  Velocities  are  affected  by  the 
viscosity  of  the  water,  variations  in  this  factor 
being  caused  by  differences  hi  temperature  and 
by  impurities  in  solution  and  in  suspension. 
The  transportation  of  small  particles  is  affected 
by  adhesion,  a  property  varying  with  the  min- 
eral character  of  the  particles  and  with  the 
impurities  of  the  water.  These  factors  were, 
ignored  but  are  probably  negligible  in  com- 
parison with  the  factors  tested.  It  may  be 
mentioned,  however,  that  the  water  of  the 
laboratory  was  practically  free  from  sus- 
pended material,  whereas  that  of  rivers  is  usu- 
ally highly  charged  at  the  time  of  most  active 
traction. 

These  comparisons  serve  to  show  that  the 
investigation  treats  of  a  group  of  important 
factors  of  the  general  problem  of  stream 
traction  but  by  no  means  comprehends  all. 
Its  results  constitute  only  a  contribution  to 
the  subject. 

ACCESSORY    STUDIES. 

Incidental  and  accessory  to  the  main  in- 
quiry were  a  number  of  minor  inquiries.  One 
pertained  to  the  Pitot  tube,  a  second  to  other 
methods  of  measuring  velocity  near  the  bot- 
tom, a  third  to  the  relation  between  the  mean 
velocity  of  a  loadless  stream  and  the  rough- 
ness of  its  channel  bed,  and  a  fourth  to  the 
.  mechanical  process  of  hydraulic  traction. 

FLUME    TRACTION. 

In  the  experiments  on  flume  traction  the  bed 
of  the  channel  was  not  composed  of  loose 
d4bris  but  was  the  unyielding  bottom  of  the 
trough.  The  same  apparatus  was  used,  with 
appropriate  modifications.  In  each  experi- 
ment slope  of  channel  was  predetermined,  the 
trough  being  placed  with  definite  inclination. 
The  bed  of  the  channel  was  given  a  definite 
quality  of  roughness  or  smoothness,  and  the 
material  of  the  load  was  of  a  particular  fine- 
ness or  of  a  definite  mixture  of  sizes.  With  a 
definite  discharge  flowing  through  the  trough, 
debris  was  fed  to  the  current  at  a  definite  rate, 
and  the  rate  was  gradually  increased  until 
clogging  occurred.  The  rate  of  feed  just  be- 
fore clogging  was  then  recorded  as  the  maxi- 
mum load  under  the  particular  conditions. 
The  series  of  experiments  used  two  widths  of 


THE   OBSERVATIONS. 


19 


channel,  five  textures  of  channel  bed,  six  dis- 
charges, and  seven  grades  of  sand  and  gravel, 
besides  mixtures.  There  were  nearly  300  de- 
terminations of  load. 

APPARATUS  AND  MATERIAL. 

EXPERIMENT    TROUGHS. 

The  trough  in  which  most  of  the  experiments 
were  made  was  of  wood,  31.5  feet  long,  with  an 
inside  width  of  1.96  feet.  The  height  of  the 
sides  was  1 .8  feet  at  the  head  and  1  foot  at  the 
end,  the  change  being  made  by  a  series  of  steps. 
Its  proportions  and  general  relations  are  illus- 
trated by  figure  1.  The  surfaces  were  planed 
and  painted.  At  the  head,  where  water 
entered,  the  trough  was  connected  with  a 
tank  by  a  flexible  joint,  a  groove  of  the  under 
side  of  the  trough  bottom  resting  on  a*  semi- 
cylindric  member  of  the  tank,  so  as  to  consti- 


tute a  hinge,  and  the  walls  of  the  trough  being 
connected  with  the  sides  of  the  tank  by  a  sheet 
of  flexible  rubber.  Here  also  was  a  gate  by 
which  the  flow  from  the  tank  could  be  stopped. 
Close  to  the  opposite  end  of  the  trough  was  a 
cross  trough  1 1  feet  long,  2.5  feet  wide,  and  3 
feet  deep,  rigidly  attached  to  the  experiment 
trough  and  extending  below  it.  A  rectangular 
opening  in  the  bottom  of  the  experiment 
trough,  an  opening  having  the  width  of  that 
trough,  permitted  sand  in  transportation  by 
the  current  to  sink  into  the  cross  trough,  which 
contained  boxes  to  receive  it.  The  width  of 
the  experiment  trough  was  varied  by  means 
of  a  longitudinal  partition  which  was  given 
various  positions.  Its  width  at  the  end  was 
also  varied  by  means  of  two  oblique  partitions, 
the  "outfall  contractor,"  which  merged  with 
the  sides  a  few  feet  from  the  end  and  could  be 
adjusted  as  desired.  For  certain  experiments 


FIGURE  1.— Diagrammatic  view  o(  shorter  experiment  trough,  showing  relations  to  stilling  tank  (A),  cross  tank  (jB),  and  settling  tank  (C). 


false  bottoms  were  added,  with  surfaces  spe- 
cially prepared  as  to  roughness.  These  will  be 
specifically  described  in  connection  with  the 
corresponding  observations.  A  second  trough, 
having  the  same  function  as  the  one  just  de- 
scribed was  150  feet  long  but  similar  in  width 
and  style.  Its  sides  were  higher  and  it  was 
not  hinged  at  the  head  but  remained  horizon- 
tal. By  temporary  arrangements  of  parti- 
tions curved  and  crooked  channels  1  foot 
wide  were  constructed  within  this  trough. 
The  shorter  trough  was  installed  in  the  base- 
ment of  the  Mining  Building  of  the  University 
of  California;  the  longer  one  on  the  campus 
near  by.  (See  PI.  I,  frontispiece.)  The  longer 
trough  was  remodeled  for  the  experiments  on 
flume  traction. 

A  third  trough  14  feet  long  and  0.67  foot 
wide  had  its  wooden  sides  replaced  for  a  space 
of  3.5  feet,  at  midlength,  by  plate  glass,  so 
that  observation  could  be  made  from  the  side. 
It  was  provided  with  a  sliding  diaphragm, 
to  be  described  in  another  place. 


A  fourth  trough,  of  iron,  was  used  only  in  the 
experiments  on  flume  traction  and  will  be 
described  hi  connection  with  those  experiments. 

A  few  experiments  were  made  also  in  a 
trough  carrying  the  waste  water  of  the  150-foot 
trough.  This  had  a  width  of  0.915  foot. 

WATER    SUPPLY. 

The  water  was  taken  from  the  municipal 
mains  of  Berkeley.  As  it  was  not  practicable 
to  draw  freely  on  this  source,  a  moderate 
supply  was  made  to  serve  for  a  long  series  of 
experiments,  being  stored  in  a  sump  and 
pumped  up  as  required.  By  repeated  use  it 
acquired  a  certain  amount  of  fine  detritus  in 
suspension,  but  the  quantity  was  not  sufficient 
to  obstruct  the  view  of  the  experiments — or 
rather,  when  it  was  found  obstructive,  a  fresh 
supply  of  clear  water  was  substituted. 

THE    WATER    CIRCUIT. 

Starting  from  the  storage  tank  or  sump, 
the  water  was  lifted  by  a  power  pump  to  a 


20 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


high  trough  13  feet  above.  In  passing  through 
this  trough  it  was  first  quieted  by  baffles 
and  then  regulated  as  to  surface  level  by 
means  of  a  spillway  about  13  feet  wide,  the 
overflow  returning  directly  to  the  sump.  At 
the  end  it  sank  slowly  through  a  vertical 
shaft,  or  leg,  whence  it  issued  in  a  jet  through 
an  aperture  regulated  by  a  measuring  gate. 
After  spending  its  force  against  a  water  cushion 
it  passed  through  a  stilling  tank  and  then 
through  the  experiment  trough.  From  that 
it  fell  a  short  distance  to  a  settling  tank,  and 
thence  returned  to  the  sump. 

In  figure  2  the  circuit  is  shown  diagram- 
matically  but  without  accuracy  as  to  the 
arrangement  and  relative  sizes  of  the  parts  of 
the  apparatus. 


Baffles 


DISCHARGE. 

For  the  control  and  determination  of  the 
discharges  used  in  the  experiments,  a  measur- 
ing gate  was  provided.  Near  the  lower  end 
of  the  vertical  leg  of  the  high  trough  the  water 
issued  through  a  rectangular  opening  in  a 
brass  plate.  The  gate,  also  of  brass,  sliding 
along  the  plate,  controlled  the  size  of  the 
opening,  its  motion  being  given  by  rack  and 
pinion  and  its  position  shown  by  a  suitable 
scale.  The  head  was  about  6  feet  and  was 
determinate.  The  gate  and  its  calibration 
are  described  in  Appendix  B  (pp.  257-259). 

The  head  was  regulated  by  means  of  the 
spillway  in  the  high  trough,  and  the  amount  of 
overflow  on  the  spillway  was  controlled  by  a 


FIGURE  2.— Diagram  of  water  circuit. 


gate  valve  in  the  supply  pipe  just  above  the 
pump.  As  a  check  on  the  control,  the  posi- 
tion of  the  water  surface  was  shown  in  an 
inclined  glass  tube  outside  of  the  high  trough. 
The  index  tube  being  nearly  horizontal,  its 
meniscus  had  a  magnified  motion  and  the 
condition  of  the  head  could  be  seen  at  a 
glance.  (See  fig.  88,  p.  257.) 

SAND    FEED. 

Above  the  experiment  trough,  and  near  its 
head,  was  a  hopper-shaped  box  from  which 
sand  was  delivered  to  the  current  in  the  trough. 
The  box  ended  downward  in  an  edge  which 
stood  transverse  to  the  trough.  Along  this 
edge  were  a  series  of  openings  whose  size  was 
determined  by  a  movable  notched  plate  of 
brass.  Water  was  supplied  to  the  sand  in  the 
hopper,  both  at  the  top  and  near  the  bottom, 


the  amount  being  regulated  by  valves.  This 
water  came  from  a  small  reservoir  that  was 
kept  full  by  diverting  part  of  the  jet  issuing 
from  the  measuring  gate,  and  its  use  therefore 
added  nothing  to  the  measured  discharge. 

For  some  of  the  experiments  debris  was  fed 
by  hand,  the  quantity  being  regulated  by 
means  of  a  measuring  box  and  a  watch. 

SAND    ARRESTER. 

The  cross  trough  attached  to  the  experiment 
trough  and  extending  below  it  (see  figs.  1  and  2) 
had  along  its  bottom  a  track  on  which  moved  a 
platform  car.  This  car  carried  two  iron  boxes 
to  receive  the  sand.  The  boxes  were  rectan- 
gular and  a  little  broader  than  the  experiment 
trough.  Openings  protected  by  wire  gauze  per- 
mitted water  to  drain  from  them  when  they 
were  lifted  out. 


THE   OBSERVATIONS. 


21 


SETTLING   TANK. 

The  function  of  the  settling  tank  was  to 
catch  sand  which  was  carried  past  the  cross 
trough.  It  was  fitted  with  a  system  of  parti- 
tions providing  two  alternative  courses  through 
which  the  stream  could  be  turned,  and  with  a 
hinged  partition — the  "deflector" — by  which 
the  diversion  was  made. 

GAGE    FOR   DEPTH   MEASUREMENT. 

A  frame  resting  on  the  experiment  trough 
bore  in  vertical  position  a  slender  brass  rod. 
This  was  raised  and  lowered  by  rack  and 
pinion,  and  its  relative  height  could  be  read  on 
a  scale.  Depth  of  water  was  measured  by 
reading  the  scale  first  with  the  rod  end  at  the 
water  surface  and  again  with  it  at  the  d6bris  ' 
surface. 

LEVEL  FOR  SLOPE  MEASUREMENT. 

A  surveyors'  level  stood  a  few  yards  from  the 
trough,  about  equally  distant  from  the  ends, 
and  was  used,  with  a  light  rod,  to  measure 
relative  heights  for  the  determination  of  slopes 
of  water  surface  and  sand  surface. 

PITOT-DARCY   GAGE. 

A  pressure-gaging  apparatus  of  the  Pitot- 
Darcy  type,  but  of  special  pattern,  was  used 
to  measure  velocities  of  current.  Its  two  aper- 
tures were  directed  severally  upstream  and 
downstream.  Its  external  form  was  designed 
to  give  the  least  possible  resistance  to  the  cur- 
rent. The  reading  scale,  with  rubber-tube 
connection,  had  a  fixed  position,  while  the 
receiving  member  was  moved  from  point  to 
point.  A  fuller  description  is  contained  in 
Appendix  A  (pp.  251-256). 

SAND   ANP    GRAVEL. 

The  d6bris  used  in  the  experiments  was 
obtained  from  three  streams — Sacramento 
River  7  miles  below  the  mouth  of  the  American, 
American  River  8  miles  above  its  mouth,  and 
Strawberry  Creek  in  Berkeley.  The  de"bris 
from  the  creek  was  relatively  coarse  and  was 
used  only  in  the  experiments  on  flume  trac- 
tion. The  mean  density  of  the  river  material 
was  2.69;  that  of  the  creek  gravel  2.53.  The 
forms  of  the  grains  of  sand  are  shown  in  Plate 
II.  To  prepare  the  debris  for  use  it  was  sorted 
into  grades  by  a  system  of  sieves,  and  in  the 
laboratory  records  each  grade  was  designated 
by  the  limiting  sieve  numbers.  Thus  the 


grains  of  the  40-50  grade  passed  through  a 
sieve  with  40  meshes  to  the  inch  and  were 
caught  by  a  sieve  with  50  meshes  to  the  inch. 
For  the  sake  of  brevity  the  grades  are  com- 
monly indicated  in  this  report  by  letters  in 
parentheses — (A),  (B),  etc. — and  the  same 
notation  is  extended  to  mixtures  of  two  or  more 
sizes.  Neither  of  these  notations,  however,  is 
suited  for  the  mathematical  discussion  of  the 
laboratory  data,  and  three  others  were  devised. 
These  are,  first,  the  mean  diameter  of  particles, 
designated  by  D;  second,  the  reciprocal  of  the 
mean  diameter,  or  the  number  of  particles,  side 
by  side,  in  a  row  1  foot  long,  designated  by  F; 
third,  the  number  of  particles  necessary  to 
occupy,  without  voids,  the  space  of  1  cubic 
foot,  designated  by  Ft.  In  the  sense  that  the 
notation  of  D  distinguishes  by  magnitudes,  the 
notations  of  F  and  F2  distinguish  by  mini- 
tudes.  F  is  otherwise  called  linear  fineness, 
and  F2  bulk  fineness. 

To  determine  the  several  constants  for  a 
grade  of  debris,  a  sample  was  weighed  and  its 
particles  were  counted.  Then,  N  being  the 
number  of  particles  in  the  sample,  W  their 
weight,  G  their  density,  and  Wo  the  weight  of 
a  cubic  foot  of  water, 

„  _  NOW* 
W 

Defining  mean  diameter  as  the  diameter  of  a 
sphere  having  the  volume  of  the  average 
particle — 

»/*, 


In  the  following  table  the  grades  of  sand  and 
gravel  are  characterized  by  the  several  nota- 
tions. 

TABLE  1. — Grades  of  debris. 


Grade 
name. 

Sieves 
used  in 
separa- 
tion 
(meshes 
to  linen). 

D,  mean 
diame- 
ter of 

particles 

(root). 

F,  num- 
ber of 
particles 
to  linear 
foot. 

Ft,  number  of 
particles  to 
cubic  foot. 

Range 
olD 
or  F. 

Range 
of  F,. 

(A). 

50-60 

o.ooioo 

1,002 

1,910.000.000 

.13 

1.44 

(B  . 

40-50 

.00123 

g!2 

1,023,  000.  (XX) 

.17 

1.60 

(C).. 

30-40 

.00166 

602 

417,000.000 

.44 

2.99 

(D). 

20-30 

.00258 

388 

111.500.000 

.56 

3.80 

(E). 

10-20 

.00561 

178 

10.770.000 

.95 

7.41 

(F). 

6-  8 

.0104 

95.9 

1,685.000 

.40 

2.74 

(G). 

4-6 

.0162 

61.8 

451,000 

.43 

2.92 

(H) 

3-  4 

.0230 

43.4 

156,000 

.36 

2.51 

(I).. 

1-  2 

.0547 

[18.3] 

11,900 

[2.00] 

i*:  <»>] 

15". 

<K). 

11 

.110 
.200 

[9.1) 
5.0] 

1,440 
239 

[2.00] 
1.50] 

[8.  00) 
3.37) 

22 


TRANSPORTATION    OF    DEBRIS    BY   RUNNING    WATER. 


The  sixth  column  of  the  table  contains  an 
index  of  the  range  in  diameter  within  each 
grade,  and  this  is  also  the  range  in  linear  fine- 
ness. The  index  is  the  ratio  between  the 
diameters  of  the  apertures  of  the  two  sieves  by 
means  of  which  the  grade  was  separated.  It 
would  express  accurately  the  ratio  of  the 
diameter  of  the  largest  particle  to  that  of  the 
smallest  particle  in  the  same  grade  if  all  the 
particles  were  spherical,  or  if  all  had  precisely 
the  same  shape.  But  there  are  actual  differ- 
ences of  form  sufficient  to  modify  materially  the 
character  of  the  separation.  For  the  same 
mean  diameter  a  prolate  form  will  pass  a  sieve 
that  will  arrest  a  sphere,  and  an  oblate  form 
may  bo  arrested  where  a  sphere  will  pass.  By 
reason  of  this  qualifying  condition,  the  actual 
range  is  somewhat  greater  than  the  tabulated 
estimate.  The  range  for  bulk  fineness,  given  in 
the  last  column,  is  the  cube  of  the  range  of 
linear  fineness. 

METHODS  OF  EXPERIMENTATION. 

t 

The  methods  here  described  are  those  used  in 
the  investigation  of  stream  traction.  Those 
used  in  studying  flume  traction  are  set  forth  in 
the  chapter  on  that  subject. 

GENERAL    PROCEDURE     FOR     A    SINGLE     EXPERI- 
MENT. 

The  experiment  trough  stands  horizontal. 
The  width  of  channel  has  been  fixed  by  the 
placing  or  the  omission  of  the  partition.  The 
head  gate  is  open.  The  openings  in  the  hopper 
have  been  set  to  a  particular  size.  The  outfall 
contractor  has  been  adjusted  to  a  width  pre- 
viously found  suitable  for  the  conditions  of  the 
experiment.  The  two  sand-catching  boxes 
stand  on  the  car,  one  of  them  being  in  position 
under  the  opening  in  the  trough  bottom. 

The  pump  is  started.  The  measuring  gate  is 
opened  until  its  index  reaches  the  point  corre- 
sponding to  the  desired  discharge.  A  valve 
associated  with  the  pump  is  turned,  if  neces- 
sary, to  adjust  the  water  level  in  the  high 
trough.  When  the  flow  in  the  experiment 
trough  has  become  steady,  or  nearly  steady,  the 
sand  feed  is  started  by  opening  the  valves 
which  admit  water  to  the  hopper. 

The  "run"  has  now  begun,  and  it  is  con- 
tinued without  change  until  the  slope  of  the 
deposit  constituting  the  channel  bed  has 


become  stable.  The  car  in  the  cross  trough  is 
now  moved  so  as  to  bring  the  reserve  sandbox 
into  position,  a  stop  watch  is  started,  and  the 
deflector  in  the  settling  tank  is  shifted.  These 
changes  begin  the  period  of  load  measurement. 
During  this  period  measurements  are  made  of 
the  depth  of  the  current  and,  under  certain 
conditions,  of  the  slope  of  the  water  surface, 
and  the  character  of  the  sand  bed  is  recorded. 
The  measuring  period  is  terminated  by  again 
shifting  the  sand  boxes  and  the  deflector,  and 
the  watch  is  stopped.  The  head  gate  is  now 
closed,  sand  feed  and  pump  are  stopped,  and 
the  discharge-measuring  gate  is  closed.  Noxt 
the  slope  of  the  channel  bed  is  measured,  and 
the  sand  caught  during  the  period  recorded  by 
the  watch  is  weighed. 

DETAILS    OF    PROCEDURE. 

In  the  following  paragraphs  some  details  and 
variants  will  be  described,  with  comments  on 
methods  and  apparatus.  As  the  investigation 
had  no  precedents  to  follow,  its  methods  were 
necessarily  developed  by  a  process  of  trial  and 
and  failure.  Many  defects  of  method  were 
remedied  as  the  work  went  on.  A  few  that  were 
recognized  after  much  work  had  been  done  were 
tolerated  to  the  end,  because  their  correction 
would  have  made  the  body  of  results  less  homo- 
geneous. 

WIDTH  OF  CHANNEL. 

The  full  width  of  the  31-foot  trough  and  the 
150-foot  trough  was  1.96  feet.  By  the  use  of 
partitions  the  channel  was  given  widths  ap- 
proximately two-thirds,  one-half,  one-third, 
two-ninths,  and  one-ninth  of  a  full  width.  The 
scale  of  width  was,  in  feet,  0.23,  0.44,  0.66,  1.00, 
1.32,  1.96.  A  few  experiments  employed  also 
the  widths  1.2,  1.4,  1.6,  and  1.8  feet. 

DISCHARGE. 

As  the  discharge  was  determined  by  flow 
through  an  aperture  of  adjustable  size,  under  an 
adjustable  head,  the  precision  of  its  measure- 
ment involved  (1)  the  precision  of  setting  the 
gate  at  the  aperture,  (2)  the  precision  of  cali- 
brating the  aperture  for  different  positions  of 
the  gate,  and  (3)  the  precision  of  adjusting  the 
head. 

The  rating  (described  in  Appendix  B)  was 
eiTected  by  a  volumetric  method,  believed  to  be 
adequate,  the  average  probable  error  of  its 
determinations  being  about  1  in  500. 


U.  S.  GEOLOGICAL  SURVEY 


PROFESSIONAL  PAPER  86     PLATE  II 


>S.<£* 


•  ,£*•£.• 


^ipP 

*-,=-. ; .* c J;»T  S.J3, 


DEBRIS    USED    IN    EXPERIMENTS 


THE   OBSERVATIONS. 


23 


The  gate  was  set  by  bringing  an  index  mark 
opposite  a  graduation  mark  on  a  scale,  the  two 
marks  being  on  brass  plates  in  contact.  The 
gate  was  controlled  by  rack  and  pinion,  and 
considerable  force  was  necessary  to  move  it. 
The  limit  of  error  may  have  been  0.002  foot. 
The  ordinary  error  is  believed  to  have  been  less 
than  0.001  foot.  An  error  of  0.001  foot  in  the 
sotting  would  cause  an  error  of  0.002  ft.3/sec.,  or 
•2-J-jr  of  the  medium  discharge. 

The  determination  of  head  is  subject  to  an 
accidental  error  and  a  systematic  error.  The 
accidental  error  pertains  to  the  adjustment  of 
water  level  in  the  high  trough,  by  means  of  the 
valve  at  the  pump,  with  observation  of  the 
tube  index  of  water  level.  It  was  possible  to 
give  this  adjustment  a  refinement  comparable 
with  that  of  the  hook  gage,  but  in  practice  that 
refinement  was  not  attained,  because  a  close 
watch  was  not  kept  on  the  index.  It  was 
found  by  experience  that  the  fluctuations  of 
level  (occasioned  by  fluctuations  of  the  electric 
current  supplying  power  to  the  pump)  were 
small,  and  they  were  usually  neglected,  a  prac- 
tical calibration  of  the  valve  at  the  pump  being 
arranged  so  that  it  could  receive  the  proper 
setting  for  each  setting  of  the  discharge  meas- 
uring gate.  The  ordinary  error  of  the  adjust- 
ment of  the  head  is  estimated  at  0.003  foot, 
which  would  occasion  an  error  in  the  discharge 
of  1  in  4,000. 

The  remaining  possibility  of  error  is  con- 
nected with  the  history  of  the  apparatus.  At 
the  time  of  the  calibration  of  the  measuring 
gate  the  laboratory  occupied  temporary  quar- 
ters. In  its  removal  to  permanent  quarters 
there  was  a  measurement  and  readjustment  of 
the  vertical  distance  constituting  the  head. 
Also,  for  the  work  with  the  long  trough  the 
measuring  gate  was  transferred  to  a  replica 
of  the  high  trough,  which  may  have  differed 
in  some  particular  affecting  the  constants.  As 
the  work  of  calibration  was  at  no  time  re- 
peated, there  was  no  check  on  the  errors  which 
may  have  been  thus  introduced.  In  a  gen- 
eral way,  they  are  probably  of  the  same  order 
of  magnitude  as  the  errors  of  adjustment 
of  water  surface.  It  is  believed  that  all  other 
errors  affecting  discharge  are  small  in  compari- 
son with  that  connected  with  the  measuring 
gate. 

The  vertical  width  of  the  aperture  by  which 
discharge  was  regulated  was  2  inches.  The 


head,  measured  from  the  middle  of  the  aper- 
ture, was  6.0  feet.  The  horizontal  dimensions 
of  the  aperture,  during  experimental  work, 
ranged  from  0.1  inch  to  6.0  inches,  and  the  cor- 
responding discharges  are  given  in  the  follow- 
ing table : 

TABLE  2. — Gate  readings  and  corresponding  discharge*. 


Gate 
opening 
(inches). 

Discharge 
(ft.'/sec.). 

Gate 
opening 
(inches). 

Discharge 
(ft.'/wc.) 

0.1 

0.019 

.5 

0.272 

.2 

.039 

.6 

.290 

.3 

.058 

.7 

.308 

.4 

.075 

.8 

.327 

.5 

.093 

.9 

.345 

.6 

.111 

2.0 

.363 

.7 

.128 

2.5 

.454 

.8 

.146 

3.0 

.545 

.9 

.164 

3.5 

.639 

1.0 

.182 

4.0 

.734 

1.1 

.200 

4.5 

.828 

1.2 

.218 

5.0 

.923 

1.3 

.237 

5.5 

1.021 

1.4 

.255 

6.0 

1.119 

THE  FEEDING  OF  SAND. 

The  fact  that  the  hourglass  has  been  used  to 
measure  time  suggests  that  the  flow  of  dry 
sand  through  an  aperture  may  be  uniform. 
Such  a  flow  was  not  tested  in  the  laboratory 
because  the  plan  for  experimentation  required 
that  sand  should  be  used  over  and  over,  and 
it  was  not  practicable  to  dry  it.  The  hopper 
was  a  device  intended  to  produce  a  uniform 
flow  of  wet  sand.  Moist  sand  will  not  flow 
through  a  small  opening;  but  if  enough  water 
is  present  to  more  than  fill  the  voids,  adhesion 
is  overcome  and  flow  takes  place,  as  in  a  quick- 
sand. The  freedom  of  the  flow  depends  on  the 
amount  of  water.  It  was  found  difficult  to 
maintain  a  uniform  condition  in  the  hopper. 
Another  difficulty  arose  from  clogging  of  the 
openings,  and  this  was  occasioned  by  shreds  of 
wood  fiber  and  similar  impurities  in  the  sand. 
The  second  difficulty  was  largely  obviated, 
after  a  time,  by  making  the  openings  larger 
and  fewer;  but  the  hopper  feeding  was  at  best 
not  sufficiently  uniform  to  be  used  in  measur- 
ing the  load  carried  by  the  experimental 
stream. 

For  all  experiments  in  which  a  large  quantity 
of  ddbris  was  carried,  the  material  was  fed  to 
the  current  by  hand  and  was  measured  in  the 
feeding.  A  small  box  of  known  capacity  was 
filled  with  the  material  and  emptied  into  the 
current  at  regular  intervals  timed  by  a  watch 
or  clock.  If  the  interval  was  long,  the  meas- 
ured unit  was  dumped  on  a  sloping  table  above 


24 


TRANSPORTATION    OF   DEBBIS  BY   SUNNING   WATER. 


the  trough  and  gradually  fed  to  the  current  by 
means  of  a  scraper.  Hand  feeding  had  the 
defect  of  discontinuity,  as  well  as  irregularity 
in  detail,  but  it  had  the  advantage  of  measure- 
ment, and  in  certain  experiments  its  meas- 
urement of  load  gave  an  important  check  on 
the  measurement  of  debris  delivered  at  the 
outfall  end  of  the  trough.  With  a  perfect  and 
stable  adjustment  of  conditions  the  two  should 
agree,  and  their  disagreement  served  to  show 
that  the  slope  of  the  channel  bed  had  not 
become  perfectly  adjusted,  or  else  that  its 
adjusted  condition  was  subject  to  rhythmic 
oscillation. 

In  some  of  the  later  work  the  rate  of  feed 
was  measured  from  time  to  time  by  inter- 
cepting the  stream  of  sand  falling  from  the 
hopper  during  a  definite  number  of  seconds  and 
weighing  the  sample  thus  caught. 

THE  COLLECTION  OF  SAND. 

In  the  original  construction  of  the  apparatus 
for  arresting  the  sand  the  opening  in  the 
bottom  of  the  trough  was  covered  by  a  coarse 
wire  screen,  which  lay  flush  with  the  trough 
bottom.  This  was  intended  to  separate  the 
current  above  from  the  still  water  below  and 
prevent  the  formation  of  eddies,  which  might 
keep  the  sand  from  settling  to  the  collecting 
box  and  might  also  check  the  current.  It 
fulfilled  its  purpose  and  was  altogether  satis- 
factory for  currents  of  moderate  velocity,  but 
with  high  velocities  it  interfered  with  the 
arrest  of  the  sand,  letting  a  considerable 
fraction  pass  on  to  the  settling  tank.  It  was 
accordingly  removed,  apparently  without  bad 
results.  Eddies  were  formed,  but  the  antici- 
pated difficulties  were  not  realized. 

On  the  whole  the  apparatus  for  arresting 
sand  was  successful.  It  was  only  with  the 
finer  debris  and  at  the  highest  velocities  that 
the  fraction  of  load  escaping  to  the  settling 
trough  was  too  large  to  be  neglected  in  the 
weighing. 

DETERMINATION  OF  LOAD. 

The  sand  collected,  in  sand  box  and  settling 
tank,  during  the  period  recorded  by  the  stop 
watch  was  weighed  without  drying,  and  the 
gross  weight  was  afterward  corrected  by  an 
allowance  for  the  contained  water.  In  order 
to  determine  the  proper  allowance  a  prelimi- 
nary study  had  been  made,  and  as  a  result  of 


that  study  a  definite  procedure  was  adopted 
for  bringing  the  wet  sand  to  a  particular 
"standard"  condition.  After  the  sand-collect- 
ing box  had  been  lifted  from  the  trough  all 
water  which  would  drain  from  it  by  gravity 
alone  was  allowed  to  escape.  It  was  then 
removed  to  smaller  boxes  for  weighing.  These 
boxes  were  jarred  by  tapping,  which  caused 
the  sand  grains  to  readjust  their  contacts  and 
settle  together,  excluding  a  part  of  the  inter- 
stitial water,  which  appeared  at  the  surface 
and  was  poured  off.  The  sand  was  then 
weighed.  It  is  of  interest  to  note  that  in  the 
condition  thus  adopted  as  a  convenient  stand- 
ard sand  occupies  less  space  than  when  dry, 
moist,  or  supersaturated;  its  voids  are  at  a 
minimum. 

The  period  recorded  by  the  stop  watch  was 
ordinarily  about  10  minutes  but  was  made  less 
when  the  current  was  most  heavily  loaded, 
because  of  the  limited  capacity  of  the  sand- 
collecting  box,  and  was  extended  for  the 
lightest  loads.  Its  beginning  was  sharply 
defined  by  the  shifting  of  the  sand  boxes, 
which  could  be  made  to  coincide  within  a 
second  with  the  starting  of  the  watch.  Its 
end  was  somewhat  less  definite,  but  the  error 
in  time  is  believed  to  be  small  in  comparison 
with  the  whole  period. 

The  load  per  second  was  computed  by 
dividing  the  total  load,  namely,  the  corrected 
weight  of  sand,  by  the  number  of  seconds  in 
the  stop-watch  reading.  Its  error  included 
(1)  the  error  of  timing,  (2)  the  error  of  stand- 
ardizing the  sand  and  correcting  for  contained 
water,  and  (3)  the  error  in  weighing.  There 
are  no  definite  data  bearing  on  its  amount, 
and  nothing  better  can  be  recorded  than  a 
general  impression  that  the  results  are  reliable 
within  2  per  cent,  that  the  precision  is  lower 
than  that  of  the  discharge  measurement,  and 
that  the  error  in  determination  of  load  is 
notably  less  than  the  error,  presently  to  be 
considered,  in  correlating  load  with  slope. 

When  the  rate  of  feed  was  regulated  by  the 
periodic  contribution  of  a  measureful  of  d6- 
bris,  the  weighings  of  the  unit,  from  time  to 
time,  showed  inequalities  from  which  precis- 
ion could  be  estimated.  A  computation  indi- 
cated the  average  probable  error,  for  a  run,  as 
about  1  per  cent.  This  depended  chiefly  on 
the  standardization,  and  as  that  was  less  per- 
fect for  the  debris  as  fed  than  for  the  debris 


THE   OBSERVATIONS. 


25 


as  collected,  the  ordinary  measurement  of  col- 
lected load  is  presumably  affected  by  a  smaller 
probable  error. 

DETERMINATION  OF  SLOPE. 

The  observations  of  slope  were  made  with 
surveyors'  level  and  rod.  The  rod,  made  for 
the  purpose,  without  unnecessary  length  or 
weight,  was  graduated  to  hundred  ths  of  a  foot 
and  read  by  eye  estimate  to  thousandths.  It 
was  held  by  an  assistant  while  the  observer 
and  recorder  stood  at  the  telescope.  The  po- 
sitions were  determined  by  a  graduation  of  the 
trough,  which  was  marked  at  every  foot.  To 
measure  the  water  slope,  heights  of  the  sur- 
face were  taken  at  several  points  along  the 
trough.  To  measure  the  sand  slope,  heights 
were  taken  at  intervals  of  «ither  2  or  4  feet,  the 
shorter  interval  being  used  with  the  shorter 
trough.  The  water  slope  could  not  be  meas- 
ured when  the  surface  was  rough.  When  the 
debris  surface  was  rough,  it  was  usually  graded 
before  measurement  by  scraping  from  crests 
into  adjacent  hollows. 

The  observed  heights  were  plotted  on  section 
paper,  with  relatively  large  vertical  scale,  and 
a  straight  line  was  drawn  through  or  among 
them.  The  line  served  the  purpose  of  a  pre- 
liminary determination  of  slope,  and  the  plots 
were  inspected  for  the  detection  of  systematic 
errors.  As  a  result  of  this  inspection  a  portion 
of  the  profile  was  selected  for  the  determina- 
tion of  slope,  and  from  the  observations  on  this 
portion  the  slope  was  computed  by  least- 
squares  method. 

CONTRACTOR. 

As  will  be  explained  more  fully  in  another 
connection,  the  slope  measurements  were  af- 
fected (1)  by  systematic  errors  connected  with 
the  conditions  under  which  the  water  entered 
and  escaped  from  the  trough,  and  (2)  by  acci- 
dental errors  arising  from  rhythm.  One  of  the 
measures  used  to  diminish  the  systematic  er- 
rors  was  the  contraction  of  the  current  at  the 
outfall  end  of  the  trough.  The  apparatus  for 
this  purpose  consisted  of  two  boards  as  wide 
as  the  depth  of  the  trough  and  arranged  as  in 
figure  3.  Their  attachment  to  the  sides  was 
flexible,  so  that  the  degree  of  convergence  and 
the  width  of  aperture  at  the  outfall  could  be 
modified  at  will.  This  apparatus  will  be  called 


the  outfall  contractor.  The  theory  and  effi- 
ciency of  the  contractor  will  be  considered  in 
the  discussion  of  the  slope  errors. 


FIGURE  3.— The  contractor. 


MEASUREMENT  OF  DEPTH. 


The  depth  of  the  current  was  measured  ai 
mid  width  and  near  midlength  of  the  trough. 
The  determination  was  made  by  means  of  the 
gage  already  described  (p.  21),  during  the 
period  of  tune  for  which  the  load  was  measured. 
As  the  water  surface  was  subject  to  rhythmic 
fluctuation,  a  series  of  observations  of  its  posi- 
tion were  made,  and  their  mean  was  used.  A 
series  of  observations  of  the  position  of  the  de- 
bris surface  were  sometimes  made  also,  but 
usually  only  a  single  observation,  and  the  read- 
ing obtained  was  subtracted  from  the  mean  of 
readings  on  the  water  surface.  The  observa- 
tions of  the  d6bris  surface  were  subject  to  an 
error  which  was  regarded  as  more  serious  than 
that  of  the  observations  of  water  surface  be- 
cause, being  essentially  systematic,  it  could 
not  be  eliminated  by  repetition.  The  prssence 
of  the  gage  rod  in  the  water  modified  the  dis- 
tribution of  velocities,  and  this  modification  in- 
cluded an  increase  of  the  current's  velocity  a 
little  below  the  end  of  the  rod.  As  the  bot- 
tom was  approached  by  the  rod,  the  current 
scoured  a  hollow  in  the  bed  immediately  under 
it;  and  if  the  rod  were  lowered  to  actual  con- 
tact, the  reading  would  give  an  excessive  esti- 
mate of  depth.  What  was  attempted  was  to 
lower  the  rod  to  a  position  as  nearly  as  possible 
at  the  level  of  undisturbed  parts  of  the  bod 
surrounding  the  visible  hollow.  This  was  a 
matter  of  judgment,  but  not  of  confident  judg- 
ment, because  the  actual  bed  was  concealed  by 
a  cloud  of  saltatory  debris  particles.  It  is 
therefore  recognized  that  the  measurements  of 
depth  are  uncertain. 

Whenever  the  water  profile  as  well  as  the 
debris  profile  was  surveyed,  an  independent 


26 


TRANSPORTATION    OF    DEBRIS   BY   RUNNING    WATER. 


estimate  of  depth  was  obtained  by  subtracting 
one  profile  from  the  other.  This  mode  of  de- 
termination avoided  the  error  incident  to  the 
gage  work  and  was  on  the  whole  satisfactory, 
but  unfortunately  the  number  of  experiments 
to  which  it  could  be  applied  was  not  large. 
During  the  greater  part  of  the  experimentation 
the  importance  of  the  water  profile  was  not 
recognized,  and  this  particular  use  of  it  was 
essentially  an  afterthought. 

The  values  from  profiles  being  assumed  to 
have  relatively  small  errors,  both  systematic 
and  accidental,  it  is  possible  to  measure  by 
their  aid  the  precision  of  the  values  from  gage 
readings.  Of  118  depths  which  were  meas- 
ured by  both  methods,  the  gage  gave  the 
greater  value  for  36,  the  lesser  for  78 ;  and  the 
average  for  gage  values  was  0.0045  ±0.0007 
foot  less  than  the  average  for  profile  values. 
Independently  of  this  apparent  systematic  er- 
ror, the  probable  error  of  a  single  measure- 
ment with  the  gage  was  ±0.007  foot. 

MEASUREMENT  OF  VELOCITY. 

The  mean  velocity  of  the  current  is  compu- 
ted by  dividing  the  discharge  by  the  area  of 
the  cross  section,  or  the  product  of  width  and 
depth.  Its  precision  depends  on  those  of  the 
determinations  of  discharge,  width,  and  depth; 
and  as  the  precision  for  discharge  and  width 
is  relatively  very  high,  the  precision  of  mean 
velocity  may  be  regarded  as  identical  with  that 
of  depth. 

The  attempts  to  measure  velocity  close  to 
the  channel  bed  were  not  successful.  This  is 
much  regretted,  because  it  is  believed  that  bed 
velocity  is  a  prime  factor  in  traction  and  that 
slope  and  discharge  exert  their  influence  chiefly 
through  bed  velocity.  The  mode  of  measure- 
ment to  which  most  attention  was  given  was 
that  by  the  Pitot-Darcy  gage,  and  special 
forms  of  that  instrument  were  constructed  for 
the  purpose.  The  difficulty  which  seemed  in- 
superable was  essentially  the  same  as  that  en- 
countered in  the  measurement  of  depth.  As 
the  instrument  approached  the  current-molded 
bed  of  ddbris,  the  bed  retreated,  with  the  for- 
mation of  a  hollow.  In  the  presence  of  the 
instrument  the  normal  velocity  at  the  bed  did 
not  exist.  Inseparable  from  this  difficulty  is 
a  property  of  the  instrument.  When  it  is  held 
close  to  the  bottom  or  side  of  a  channel  its  con- 
stant is  not  the  same  as  in  the  free  current. 


The  system  of  flow  lines  and  velocities  with 
which  the  stream  passes  the  obstructing  object 
determines  the  instrument's  constant,  and  when 
that  system  is  modified  by  a  neighboring  ob- 
ject the  constant  changes.  The  nature  of  these 
difficulties  is  such  that  it  was  not  thought 
worth  while  to  experiment  with  other  gages 
and  meters  which  limit  the  freedom  of  the 
current. 

Other  devices  tried  were  of  one  type.  Small 
objects,  such  as  currants  or  beans,  only  slightly 
denser  than  water,  were  placed  in  the  current 
and  watched.  The  lighter  ones  would  not 
remain  near  the  bottom.  The  heavier  ones 
were  visibly  retarded  when  they  touched  the 
bed  and  were  also  retarded  when  close  to  the 
bottom  by  the  cloud  of  saltatory  sand,  which 
has  a  slower  average  velocity  than  the  water  it 
suffuses. 

MODES  OF  TRANSPORTATION. 

MOVEMENT    OF    INDIVIDUAL    PARTICLES. 
ROLLING. 

In  stream  traction  sliding  is  a  negligible  fac- 
tor. The  roughness  of  the  bed  causes  particles 
that  retain  contact  to  roll.  When,  as  in  most 
of  the  experiments,  the  grains  are  of  nearly 
uniform  size,  each  moving  grain  has  to  sur- 
mount obstacles  with  diameter  like  its  own, 
and  when  it  reaches  the  summit  of  an  obstacle 
it  usually  possesses  a  velocity  which  causes  it 
to  leap.  So  rolling  is  chiefly  the  mere  prelude 
to  saltation.  With  mixed  d4bris  the  same  is 
true  for  the  finer  grains,  but  the  coarser  may 
roll  continuously  over  a  surface  composed  of 
the  finer,  and  the  coarsest  of  all,  those  close  to 
the  limit  of  competence,  move  solely  by  rolling. 

The  large  particle,  as  it  rolls  over  the  bed  of 
smaller  particles,  indents  the  bed,  and  its  con- 
tact involves  friction.  The  energy  thus  ex- 
pended comes  from  the  motion  of  the  water, 
and  its  communication  depends  on  differential 
motion  between  water  and  particle.  Except 
under  special  conditions,  to  be  mentioned  later, 
the  load  travels  less  rapidly  than  the  carrier, 
and  it  is  also  true  that  in  a  load  of  mixed  debris 
the  finer  parts  outstrip  the  coarser. 

SALTATION. 

In  stream  traction  the  dominant  mode  of 
particle  movement  is  saltation.  Because  salta- 
tion grades  into  suspension  it  has  often  been 


THE   OBSERVATIONS. 


27 


explained  in  the  same  manner,  by  appeal  to  up- 
ward movement  of  filaments  of  current,  but 
the  recent  studies  have  led  me  to  entertain 
a  different  view.  Before  this  view  is  presented 
an  account  will  be  given  of  certain  observations 
which  were  made  with  the  use  of  the  trough 
having  glass  sides. 

Through  the  trough  was  passed  a  current 
transporting  sand  of  uniform  grade,  and  the 
conditions  were  such  that  the  sand  bed  and 
water  surface  were  smooth.  In  the  same  water 
floated  a  few  fine  particles  and  thin  flakes  of 
mica,  illustrating  suspension,  but  there  was  no 
intergradation  of  the  two  processes.  Viewed 
from  the  side,  the  saltation  was  seen  to  occupy 
at  the  bottom  of  the  current  a  space  with  a 
definite  upper  limit,  parallel  to  the  sand  bed. 
Within  this  space — the  zone  of  saltation — the 
distribution  of  flying  grains  was  systematic, 
the  cloud  being  dense  below  and  thin  above, 
but  not  perceptibly  varying  from  point  to 
point  along  the  bed.  Viewed  from  above,  the 
surface  of  the  cloud  seemed  uniform  and  level, 
and  it  all  appeared  to  be  moving  in  the  same 
direction.  There  was  no  suggestion  of  swirls 
in  the  current. 

When,  in  looking  from  the  side,  attention 
was  directed  to  the  base  of  the  zone,  it  was 
easy  to  watch  grains  that  traveled  half  by 
rolling  and  half  by  skipping,  and  these  moved 
quite  slowly;  but  higher  in  the  zone  the 
motions  were  so  rapid  and  diverse  that  all  was 
a  blur.  To  resolve  this  blur  a  sliding  dia- 
phragm was  arranged.  This  consisted  of  a 
short  board  with  a  hole  in  it.  The  board  hung 


FIGURE  4. — Diagrammatic  view  of  part  of  experiment  trough  with  glass 
panels  (A )  and  sliding  screen  (B).     C,  Hole  in  screen. 

outside  the  wall  of  the  trough,  being  supported 
by  a  cleat  above  in  such  manner  that  it  could  be 
slid  along  the  trough.  (See  fig.  4.)  The  hole, 
about  2  niches  square,  gave  a  restricted  view  of 
the  saltation  zone.  By  sliding  the  board  in  the 
direction  of  the  current  and  keeping  the  eyes 
opposite,  a  traveling  field  of  view  was  obtained. 
Manifestly  if  the  field  traveled  at  the  same  rate 
as  the  current,  any  object  moving  with  the 
current  would  appear  at  rest  to  the  observer, 
because  there  would  be  no  relative  motion  of 


observer  and  object;  and  if  objects  in  the  water 
were  moving  (horizontally)  at  different  rates, 
those  coinciding  hi  rate  with  the  field  would  be 
seen  as  if  at  rest,  while  the  others  would  be  seen 
as  moving. 

When  the  field  was  moved  slowly  the  rolling 
grains  ceased  to  be  distinct  but  were  replaced 
in  distinctness  by  grains  that  seemed  to  bob 
up  and  down.  These  vibrated  through  a  space 
of  two  or  three  diameters,  as  if  repeatedly 
striking  the  bed  and  rebounding.  In  inter- 
preting this  appearance,  allowance  must  be 


FIGURE  5. — Appearance  of  the  zone  of  saltation,  as  viewed  from  the  side 
with  a  moving  fleld. 

made  for  the  fact  that  the  grains  were  dis- 
tinctly seen  because  they  were  moving  hori- 
zontally about  as  fast  as  the  diaphragm.  Their 
paths  were  really  low-arching  curves,  and  only 
the  vertical  factor  remained  when  the  hori- 
zontal was  abstracted.  It  is  probable  also  that 
the  appearance  of  rebounding  was  largely 
illusory,  most  of  the  grams  either  stopping  at 
the  end  of  the  leap,  or  else  leaping  next  time 
with  a  different  velocity. 

When  the  field  was  moved  somewhat  faster, 
the  bobbing  grains  disappeared  and  there  came 
into  distinct  vision  a  set  of  grains  quite  free 
from  the  bed  and  occupying  a  belt  within  the  sal- 
tation zone.  All  the  zone  above  and  below  them 
was  blurred.  In  the  middle  of  the  belt  vertical 
motion  was  to  be  discerned  but  wa*  less  con- 
spicuous than  in  the  lower  zone.  Where  dis- 
tinctness graded  into  blurring,  lines  of  motion 
could  be  seen  which  were  oblique  and  curved, 
the  lines  above  the  belt  curving  forward  and 
those  below  backward,  as  shown  in  figure  5. 

With  progressively  faster  motion  of  the  field 
the  belt  of  distinct  vision  rose  higher,  until  the 
top  of  the  zone  was  reached,  when  all  the  lower 
part  was  blurred. 

The  systematic  gradation  of  velocity  and 
other  features  from  the  bed  upward  and  the 


28 


TRANSPORTATION    OF    DEBRIS   BY    RUNNING    WATER. 


sensible  uniformity  of  process  over  the  whole 
width  of  channel  are  not  consistent  with  the 
idea  that  the  saltation  zone  is  invaded  by 
eddies  of  large  dimensions,  such  as  would  bo 
competent  to  sustain  the  grains  by  the  up- 
ward components  of  their  motions.  If  there 
were  large  ascending  and  descending  strands 
of  current  the  visible  surface  of  the  zone 
would  be  locally  raised  and  depressed  by 
them.  We  must,  indeed,  assume  that  the  flow 
is  turbulent,  in  the  technical  sense,  because 
parallel  or  laminar  flow  is  impossible  with 


FIGURE  6.  —  The  beginning  of  a  leap,  in  saltation. 

velocities  competent  for  traction,  but  the  ed- 
dies may  be  assumed  to  be  of  small  dimen- 
sions in  relation  to  the  depth  of  the  zone,  and 
the  lines  of  flow  with  which  saltation  is  con- 
cerned may  be  assumed  to  be  approximately 
parallel  to  the  general  direction  of  the  current. 

The  explanation  I  would  substitute  for  that 
of  the  uplift  of  grains  by  rising  strands  is  that 
each  gram  is  projected  from  the  bed  with  an 
initial  velocity  which  gives  it  a  trajectory  an- 
alogous to  that  of  a  cannon  ball.  The  follow- 
ing fuller  statement,  though  given  with  little 
qualification,  should  be  understood  as  largely 
hypothetic. 

In  figure  6  the  current  is  supposed  to  move 
from  left  to  right  above  the  grains  of  debris 
shown  in  outline.  A  grain  which  in  A  is  at 
rest  appears  in  B  in  an  advanced  position, 
having  been  rolled  upward  and  forward  about 
an  undisturbed  grain  which  lies  in  its  way. 
(The  moving  grain  is  doubtless  more  likely  to 
roll  against  two  other  grains  than  a  single  one, 
but  the  principle  is  the  same.)  In  moving  to 
its  new  position  the  center  of  gravity  of  the 
gram  describes  a  curve  convex  upward.  The 
grain  continuously  gains  in  velocity,  and  the 
acceleration  also  increases  as  the  direction  of 
motion  comes  to  make  a  smaller  angle  with 
the  direction  of  the  current.  At  each  instant 
the  accelerative  force  due  to  the  current  and 
that  of  gravity  are  combined  and  have  a  re- 
sultant direction;  and  the  combined  or  re- 
sultant accelerative  force  may  be  resolved  into 
two  parts,  one  of  which  coincides  in  direction 


with  the  motion  of  the  center  of  gravity  and 
the  other  with  the  line  joining  the  center  of 
gravity  and  the  point  of  contact.  The  last- 
mentioned  component  presses  the  moving 
gram  against  the  stationary  grain.  Opposed 
to  it  is  the  centrifugal  force  arising  from  the 
curvature  of  the  grain's  path;  and  the  point 
is  finally  reached  where  the  centrifugal  force 
dominates  and  the  grain  is  free.  Under  the 
ordinary  conditions  of  saltation  this  point  is 
not  the  crest  of  the  obstruction,  but  is  on  the 
upstream  side,  so  that  the  grain's  direction  of 
motion  at  the  instant  of  separation  is  obliquely 
upward.  Thus  the  free  grain  is  initially  mov- 
ing upward  as  well  as  forward,  and  it  has  al- 
most literally  made  a  leap  from  the  bed. 

If  the  grain  were  at  that  instant  released 
from  all  influences  but  gravity,  its  path  before 
returning  to  the  bed  would  be  the  arc  of  a 
quadric  parabola  with  vertical  axis.  The 
actual  deviation  of  its  trajectory  from  the 
parabolic  form  is  analogous  to  that  observed 
in  gunnery,  for  it  arises  from  the  resistance  of 
a  fluid;  but  the  laws  of  resistance  are  not  the 
same  for  air  and  water,  and  the  frictional  ac- 
celeration in  one  case  is  negative  while  in  the 
other  it  is  mainly  positive.  The  trajectory  in 
gunnery  is  shorter  than  the  ideal  parabolic 
arc;  in  saltation  it  is  longer. 

Figure  8  gives  diagrammatically  the  trajec- 
tory of  a  saltatory  grain.  In  figure  7  AB 
is  a  portion  of  the  same  trajectory.  Let  the 
space  AC  represent  the  instantaneous  velocity 
of  the  grain,  and  let  the  line  AD  represent  in 
direction  and  length  the  velocity  of  the  water 


FIGURE  7. — Diagram  of  accelerations  affecting  a  saltatory  grain. 

about  the  grain.  Then,  C  and  D  being  con- 
nected by  a  line,  CD  represents  in  direction 
and  magnitude  the  relative  velocity  of  water 
and  grain,  or  the  velocity  of  the  water  as  re- 
ferred to  the  grain.  By  reason  of  the  mutual 
resistance  of  water  and  grain,  this  relative  mo- 
tion accelerates  the  grain,  the  acceleration  be- 
ing a  function  of  the  differential  velocity,  the 
size  of  the  grain,  and  other  conditions.  On  the 
line  CD,  showing  the  direction  of  the  accelera- 
tion, let  the  space  CE  represent  its  amount. 
Then  from  E  draw  the  vertical  EF,  represent- 


THE   OBSERVATIONS. 


29 


ing,  to  the  same  scale,  the  acceleration  of  the 
grain  by  gravity.  Connect  C  and  F;  the  line 
CF  represents  in  magnitude  and  direction  the 
resultant  acceleration  of  the  grain.  These 
relations  are  independent  of  the  particular 
directions  of  motion  of  the  grain  and  the  water. 
Let  us  now  introduce  the  assumptions,  believed 
to  be  practically  true  for  the  laboratory  condi- 
tions, that  the  water  in  the  region  of  saltation 
moves  parallel  to  the  bed  and  that  its  velocity 
increases  notably  with  distance  from  the  bed. 
In  the  ascending  part  of  its  path  the  grain  en- 
counters filaments  of  the  current  with  higher 
and  higher  velocity.  This  tends  to  increase 
the  relative  velocity,  but  the  grain  is  at  the 
same  time  gaining  in  horizontal  velocity  and 
the  gain  tends  to  diminish  the  relative  velocity. 
Unless  the  leap  is  short  in  relation  to  the  size 
of  the  grain,  the  second  of  these  tendencies  is 
the  greater,  and  at  the  highest  point  of  its 
path  the  grain  is  moving  nearly  as  fast  as  the 


FIGURE  8. — Theoretic  trajectory  of  a  saltatory  particle,  the  initial  point 
being  at  /.    Arrows  indicate  acceleration. 

water.  In  the  descending  part  of  its  path  it  en- 
counters slower  moving  filaments  of  current, 
and  at  some  point  (H,  fig.  8)  its  horizontal  mo- 
tion may  equal  that  of  the  adjacent  water. 
Then  beyond  H  it  passes  through  filaments 
moving  still  more  slowly,  and  its  acceleration 
from  the  reaction  of  the  current  becomes  nega- 
tive. The  acceleration  due  to  gravity  is  of 
course  uniform  and  downward,  and  its  combina- 
tion with  that  due  to  the  current  yields  a  system 
of  directions  and  magnitudes  of  the  type  indi- 
cated in  figure  8  by  short  arrows.  In  the 
shorter  and  lower  trajectories  it  is  probable 
that  the  critical  point  //is  not  reached. 

If  the  position  of  the  grain  before  leaping 
(fig.  6)  is  such  that  only  a  relatively  short  roll 
suffices  to  free  it,  then  its  initial  velocity  is 
small  and  the  angle  of  ascent  at  which  it  is 
freed  is  low.  It  has  a  short,  flat  trajectory,  and 
its  velocity  at  the  highest  point  is  moderate. 
If  the  original  roll  is  longer  there  is  time  to 
acquire  speed  before  the  leap;  the  initial  ve- 
locity is  large  and  the  angle  of  ascent  is  rela- 
tively high.  It  has  a  long  and  high  trajectory 
and  when  at  the  crest  has  been  accelerated  to 
high  velocity.  IF  a  grain  at  the  end  of  a  leap 


touches  the  bed  at  a  favorable  point  it  may  leap 
again  without  coming  to  rest,  and  the  impetus 
of  the  first  flight  will  thus  enhance  the  initial 
velocity  of  the  second. 

In  the  observations  with  the  moving  field 
the  grains  seen  most  distinctly  were  those  which 
moved  horizontally  with  the  field  and  at  the 
same  time  had  little  vertical  motion.  So  each 
belt  of  distinctness  contained  grains  at  the  tops 
of  their  trajectories  and  was  practically  made 
up  of  such  grains.  The  grains  producing  the 
curved  lines  in  figure  3  were  ascending  or  de- 
scending obliquely,  and  their  horizontal  com- 
ponents of  motion  coincided  with  the  motion  of 
the  field  for  an  instant  only. 

In  general  the  observations  seem  to  show 
that  the  summit  velocities  of  the  leaping  grains 
increase  systematically  with  the  height  of  the 
leap,  and  this  generalization  is  in  perfect  accord 
with  the  hypothesis  that  the  paths  of  grains 
are  determined  primarily  by  initial  impulse. 

Under  the  hypothesis  the  series  of  velocities 
observed  by  aid  of  the  moving  field  are  not 
velocities  of  current,  for  the  initial  velocities  of 
grains,  being  caused  by  the  current,  require  that 
the  water  outspeed  all  the  grains  at  the  bottom 
of  the  zone  of  saltation.  At  the  top  of  the 
zone  there  must  be  at  least  a  slight  advantage 
with  the  current,  provided  the  water  velocities 
increase  upward.  That  the  water  velocities  do 
increase  upward  can  hardly  be  doubted,  for  in 
sweeping  along  the  sand  the  stream  expends 
energy,  and  as  its  energy  subsists  in  velocity, 
the  expenditure  involves  retardation.  More- 
over, the  grains  of  sand  are  at  the  same  time 
most  numerous  and  slowest  near  the  bottom 
of  the  zone,  so  that  their  effect  is  there  greatest. 

In  this  connection  it  is  to  be  observed  that 
the  width  of  the  belt  of  distinct  vision  in  the 
moving  field  (fig.  5)  is  greater  for  the  upper 
part  of  the  zone  of  saltation  than  for  the  lower. 
As  distinct  vision  is  limited  to  a  certain  (unde- 
termined) range  in  horizontal  velocities,  this 
fact  implies  that  the  increase  in  horizontal 
speed  of  sand  grains  with  distance  from  the 
bed  is  less  rapid  in  the  upper  part  of  the  zone 
than  in  the  lower. 

The  preceding  discussion  is  subject  to  two 
qualifications,  the  first  of  which  is  connected 
with  the  retardation  of  the  current  at  the  side 
of  the  trough.  By  reason  of  that  retardation 
the  zone  of  saltation  is  shallower  near  the  side 
and  does  not  include  the  longer  and  higher 


30 


TRANSPORTATION    OF    DEBEIS   BY    RUNNING    WATER. 


leaps.  Figure  9  gives  an  ideal  conception  of 
the  cross  section  of  the  zone  and  the  distribu- 
tion of  flying  grains  within  it.  Observation 
from  the  side  penetrates  but  a  short  distance 
into  the  cloud,  the  distance  being  least  where 
the  cloud  is  most  dense.  The  practical  limit 
of  visual  penetration  may  be  assumed  to  take 
some  such  form  as  the  line  AB.  Thus  the 


FIGURE  9. — Ideal  transverse  section  of  zone  of  saltation  at  side  of  experi- 
ment trough. 

tract  actually  studied  in  the  work  with  the 
moving  field  was  somewhat  superficial  and  was 
not  in  strictness  a  vertical  section  of  the  zone. 
The  second  qualification  is  connected  with 
turbulence.  In  steady  flow  the  motion  at  each 
point  of  a  stream  is  constant  in  velocity  and 
direction.  When  the  general  velocity  exceeds 
a  certain  minimum,  which  for  the  streams  we 
have  to  consider  is  very  small,  the  flow  is  not 
steady,  but  involves  eddies  or  vortices,  which 
as  a  rule  move  onward  with  the  current.  In 
consequence  of  these  eddies  the  course  of  each 
particle  of  water  is  sinuous,  and  the  sinuous 
courses  interweave.  The  flow  is  then  said  to 
be  turbulent.  Usually  there  are  both  large 
and  small  eddies,  the  minute  ones  being  multi- 
tudinous. As  the  axes  of  whirling  movements 
have  all  attitudes,  the  directions  of  motion,  as 
a  rule,  have  upward  or  downward  components, 
and  the  suspension  of  particles  of  debris  is  due 
to  the  upward  components.  Particles  so  small 
that  they  can  not  come  to  rest  on  the  bottom 
are  thereby  lifted  and  relifted  and  kept  in  the 
body  of  the  water.  Under  the  conditions 
arranged  for  the  study  of  saltation  there 
appeared  to  be  no  large  eddies,  but  the  zone 
was  unquestionably  pervaded  by  small  ones, 
excited  by  the  roughness  of  the  bed  and  by  the 
differential  motions  of  water  and  leaping 
grains.  With  increasing  strength  of  current 


the  texture  of  turbulence  would  enlarge  and 
saltation  pass  into  suspension.  With  a  diver- 
sified debris,  instead  of  the  uniform  material 
actually  used,  there  would  be  phases  of  action 
in  which  the  paths  of  small  grains  were  made 
sinuous  by  turbulence,  while  those  of  larger 
grains  remained  simple  in  form.  The  trajec- 
tory of  saltation,  as  described,  may  therefore 
be  regarded  as  a  simple  type  of  path  which 
combines  in  all  proportions  with  the  sinuous 
type  of  path  characterizing  suspension. 

Through  the  entire  zone  of  saltation  motion 
is  being  communicated  to  particles  of  the  load 
by  the  water,  and  there  is  a  corresponding  loss 
of  motion  by  the  water.  That  loss  reduces  all 
the  stream's  velocities  but  makes  the  greatest 
reduction  in  the  lower  part  of  the  zone  of 
saltation.  The  loss  of  velocity  in  the  lower 
strands  reduces  their  power  to  cause  particles 
to  leap.  The  greater  the  load  the  greater  this 
reduction,  and  thus  the  quantity  of  load  is 
automatically  regulated. 

COLLECTIVE    MOVEMENT. 

In  the  experiment  used  to  illustrate  saltation 
the  collective  movement  of  the  sand  was  uni- 
form, the  conditions  of  the  experiment  having 
been  adjusted  to  that  end.  But  it  is  equallv 
possible  so  to  adjust  them  as  to  make  the 
collective  movement  rhythmic.  Uniformity  is 
in  fact  an  intermediate  phase  between  two 
rhythmic  phases,  which  are  of  contrasted  types. 
These  phases  will  be  described. 

In  another  experiment  a  bed  of  sand  was 
first  prepared  with  the  surface  level  and 
smooth.  Over  this  a  deep  stream  of  water 
was  run  with  a  current  so  gentle  that  the  bed 
was  not  disturbed.  The  strength  of  current 
was  gradually  increased  until  a  few  grains  of 
sand  began  to  move  and  then  was  kept  steady. 
Soon  it  was  seen  that  the  feeble  traction  did 
not  affect  the  whole  bed,  but  only  certain  tracts, 
and  after  a  time  a  regular  pattern  developed 
and  the  bed  exhibited  a  system  of  waves  and 
hollows.  As  the  waves  grew  the  amount  of 
transportation  increased,  showing  that,  under 
the  given  conditions,  the  undulating  surface 
was  better  adapted  to  traction  than  the  plane. 
With  such  waves  and  hollows  are  associated  a 
special  mode  of  transportation,  which  is  illus- 
trated in  figure  10.  A  current  reaching  the 
bed  at  A  follows  the  rising  slope  and  crest  of 
the  wave  to  C  and  then  shoots  free,  to  reach 


THE    OBSERVATIONS. 


31 


the  bed  again  at  D.  The  space  overleaped 
between  C  and  D  is  occupied  by  one  or  more 
slow-moving  eddies.  From  A  to  C  there  is 
traction,  the  material  being  derived  from  the 
slope  between  A  and  B.  At  C  the  debris, 
being  abandoned  by  the  current,  is  dumped, 
and  it  slides  by  gravity  down  the  slope  CE. 
So  the  upstream  face  of  the  wave  is  eroded  and 
the  downstream  face  built  out,  with  the  result 
that  the  wave,  as  a  surface  form,  travels 
downstream.  As  this  is  precisely  what  takes 
place  when  a  sand  hill  travels  under  the 
influence  of  the  wind,  the  name  of  the  eolian 
hill  has  been  borrowed,  and  the  waves  are 
called  dunes.1  In  one  of  the  narrower  troughs 
of  the  laboratory  the  dunes  formed  a  single 
line.  In  a  wider  trough  their  arrangement 
sometimes  suggested  a  double  line,  the  crests 


of  one  being  opposite  the  hollows  of  the  other, 
but  their  arrangement  continually  changed. 
On  the  bed  of  a  broad,  shallow  stream  they  are 
apt  to  have  a  subregular  imbricated  pattern.2 
In  a  deep  stream  a  single  dune  may  be  nearly 
as  broad  as  the  channel.  In  the  laboratory 
the  forms  were  inconstant,  but  the  type  was 
about  as  broad  as  long,  with  the  front  edge 
convex  downstream.  In  natural  streams  the 
dunes  show  great  variety  in  outline,  some  being 
described  as  longest  in  the  direction  of  the 
current  and  others  as  greatly  extended  in  the 
transverse  direction.  They  vary  in  size  with 
the  size  of  the  stream,  but  especially  with  the 
depth,  and  are  transformed  and  remodeled 
with  increase  and  reduction  of  discharge.  The 
horizontal  dimensions  of  most  laboratory 
examples  may  be  conveniently  described  in 


J): 


FIGURE  10.— Longitudinal  section  illustrating  the  dune  mode  of  traction. 


inches,  but  river  examples  may  require  scores 
or  hundreds  of  feet.3  The  maximum  height  in 
the  laboratory  was  probably  2  inches;  for 
those  in  Sacramento  River  2  feet  has  been 
reported,  and  for  those  in  the  Mississippi  22 
feet. 

In  each  series  of  laboratory  experiments  to 
determine  the  relation  of  load  to  slope  the 
initial  run  was  made  with  a  small  load,  while 
for  the  succeeding  runs  the  load  was  pro- 
gressively increased.  Enlargement  of  the  load 
caused  increase  of  slope  and  velocity,  with 
decrease  of  depth,  and  these  changes  were 
accompanied  by  changes  in  the  mode  of  trans- 
portation. In  the  earlier  runs  dunes  were 
formed,  and  these  marched  slowly  down  the 
trough.  Then,  somewhat  abruptly,  the  dunes 
ceased  to  appear,  and  for  a  number  of  runs  the 

i  This  is  the  name  chiefly  used  by  Swiss  investigators  (see  De  Candolle, 
Arch.  sci.  phys.  et  nat.,  vol.  9,  p.  242, 1883,  and  Forel,  idem,  vol.  10,  p.  43, 
1884),  and  many  observers  compare  the  subaqueous  feature  to  the  eolian; 
but  the  specific  title  commonly  used  in  the  United  States  and  Great 
Britain  is  sand  wane,  and  some  French  engineers  employ  grbee.  In  the 
present  paper,  dune  is  preferred  to  sand  wave  because  there  is  occasion 
to  distinguish  two  species  of  debris  waves. 


channel  bed  was  without  waves  and  approxi- 
mately plane,  although  somewhat  ruffled  in  the 
run  immediately  following  the  disappearance  of 
dunes.  Finally  a  third  stage  was  reached  in 
which  the  bed  was  characterized  by  waves  of 
another  type.  These  are  called  antidunes, 
because  they  are  contrasted  with  dunes  in 
their  direction  of  movement;  they  travel 
against  the  current  instead  of  with  it.  Their 
downstream  slopes  are  eroded  and  their  up- 
stream slopes  receive  deposit.  They  travel 
much  faster  than  the  dunes,  and  their  profiles 
are  more  symmetric.  The  water  surface, 
which  shows  only  slight  undulation  in  connec- 
tion with  dunes,  follows  the  profiles  of  anti- 

*  The  imbricated  pattern  is  frequently  seen  beneath  tidal  waters, 
where  ripple  marks  due  to  the  reaction  of  wind  waves  are  transformed 
into  dunes  when  the  tidal  current  sweeps  across  them.  It  is  then  usually 
to  be  ascribed  to  a  difference  in  direction  of  the  two  actions.  An  elabo- 
rate account  of  its  development  in  rivers  is  given  by  H.  Blasius,  who  has 
recently  investigated  the  whole  subject  of  the  rhythmic  features  of  river 
beds.  See  Zeitschr.  Bauwesen,  vol.  60,  pp.  465-472, 1910. 

»  Arthur  Hider,  who  studied  dunes  in  the  lower  Mississippi,  reported 
a  maximum  length,  crest  to  crest,  of  750  feet,  a  maximum  height  of  22 
feet,  and  a  maximum  progression  of  81  feet  in  a  day.  See  Mississippi 
River  Comm.  Rept.,  1882,  pp.  83-88  (=Chief  Eng.  U.  S.  A.,  Kept.,  1883, 
pp.  2194-2199). 


32 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER, 


dunes  closely  and  shares  their  transformations.1 
(See  fig.  11.)  Usually  each  antidune  occupied 
the  full  width  of  the  experiment  trough;  and  in 
natural  streams,  so  far  as  I  have  observed,  they 
either  reach  from  side  to  side  of  the  channel  or 
else  form  well-defined  rows  in  the  direction  of 
the  current.  Not  only  is  a  row  of  antidunes  a 
rhythm  in  itself,  but  it  goes  through  a  rhythmic 
fluctuation  in  activity,  either  oscillating  about 
a  mean  condition  or  else  developing  paroxys- 
mally  on  a  plane  stream  bed  and  then  slowly 
declining.  Paroxysmal  increase  starts  at  the 
downstream  end  of  a  row  and  travels  upstream, 
gaining  in  force  for  a  time,  and  the  climax  is 
accompanied  by  a  combing  of  wave  crests. 
Where  the  debris  is  very  coarse,  as  on  the  out- 
wash  plains  of  glaciers,  a  din  of  clashing  bowl- 
ders is  added  to  the  roar  of  the  water.2 

Of  the  phases  of  process  in  the  laboratory 
Mr.  Murphy  writes : 

Their  [the  dunes']  form  is  continually  changing  as  they 
move  forward;  they  divide  and  again  unite,  the  parts 
traveling  at  different  rates,  and  new  ones  form  on  top  of 

rl.O 


the  older  ones.  The  grains  roll  up  the  gentle  slope,  fall 
over  the  crest,  are  covered  by  other  grains,  and  rest  until 
the  dune  again  passes  over  them  an  d  they  are  again  uncov- 
ered. Thus  the  time  during  which  they  are  in  motion  is 
small  compared  to  the  time  during  which  they  rest.  As 
the  velocity  of  the  current  increases,  the  rate  of  feeding 
being  correspondingly  increased,  the  size  of  the  dunes  and 
their  rate  of  movement  increase.  Thus  we  find  that  when 
the  discharge  is  0.363  ft.3/sec.,  load  11  gin. /sec.,  and  slope 
0.32  per  cent,  the  dunes  are  7  to  9  inches  long  and  one-half 
inch  high  and  move  at  the  rate  of  0.56  foot  a  minute,  but 
when  the  discharge  is  0.734  ft.3/see.  and  the  load  30 
gm./sec.,  the  dunes  are  13  to  15  inches  long,  three-fourths 
of  an  inch  high  and  move  at  the  rate  of  1.5  feet  a  minute. 
As  the  velocity  of  the  current  increases  some  of  the  grains 
leap  as  well  as  roll,  and  some,  instead  of  dropping  over 
the  crest  of  a  dune  and  resting,  leap  to  the  next  dune. 
The  dune  grows  less  distinct  in  form  and  finally  at  a  criti- 
cal velocity  it  disappears,  dune  motion  ceases,  and  the 
sand  surface  becomes  comparatively  even.  This  condi- 
tion of  even  surface  flow  continues  as  the  slope  increases 
until  at  another  critical  velocity  antidune  movement 
begins.  A  profile  of  the  sand  surface  for  this  kind  of 
motion  is  shown  in  figure  11.  For  this  experiment  the 
trough  width  is  1.32  feet,  discharge  0.734  ft.3/sec.,  load 
213  gm./sec.  and  the  sand  slope  1.23  per  cent.  These  sand 
waves  are  from  2  to  3  feet  in  length  from  crest  to  crest,  they 
extend  the  width  of  the  trough,  and  some  of  them  are  0.5 


6  '9 

FIGURE  11. — Longitudinal  section  illustrating  the  antidune  mode  of  traction. 

trough. 


12  15 

The  numbers  show  distance  in  feet  from  the  head  of  the  experiment 


foot  in  height  from  crest  to  trough  of  the  wave.  They 
travel  slowly  upstream,  some  of  the  sand  being  scoured 
from  the  downstream  face  in  the  vicinity  of  Y  (fig.  11) 
and  deposited  on  the  upstream  face  at  X.  Some  of  these 
waves  remain  for  two  minutes  or  longer,  but  most  of  them 
not  longer  than  one  minute.  A  whitecap  forms  on  the 
surface  of  the  water  when  the  larger  waves  disappear. 
Sometimes  two  or  more  will  disappear  at  once  and  leave 
the  surface  without  waves  for  a  distance  of  10  feet  or  more. 
Only  a  portion  of  the  sand  transported  takes  part  in  the 
formation  of  these  sand  waves.  The  velocity  in  a  wave 
trough  is  greater  than  near  the  crest.  The  sand  grains 
flow  nearly  parallel  to  the  bed  as  they  pass  through  the 

1  Antidunes,  though  less  common  than  dunes,  are  by  no  means  rare 
under  natural  conditions.     They  are  described  by  Vaiighan  Cornish 
(Geog.  Jour.  (London),  vol.  13,  p.  624, 1899;  Scottish  Geog.  Mag.,  vol.  17, 
pp.  1-2, 1901),  and  are  mentioned  by  John  S.  Owens  (Geog.  Jour., vol.  31, 
p.  424, 1908). 

2  The  sequence  of  bed  characters — dune,  smooth,   antidune— was 
observed  by  John  S.  Owens  in  studies  with  natural  currents  in  1907,  and 
the  characters  were  correlated  with  velocities.    With  depths  of  3  to  6 
inches  and  a  bed  of  sand,  he  noted  sand  ripples  [dunes)  when  the  ve- 
locities, measured  by  floats,  were  from  0.85  to  2.5  ft./sec.,  and  the 
appearance  of  antidunes  at  a  velocity  of  about  3  ft./sec.    (Geog.  Jour. 
(London),  vol.  31,  pp.  416,  424,  1908).     Sainjon  and  Partiot,  study- 
ing the  movement  of  debris  in  the  Loire,  had  previously  observed 
that  whereas  with  low  velocities  the  entire  bottom  load  was  transported 
through  the  progress  of  dunes,  with  higher  velocities  the  del>ris  was  swept 
along  from  crest  to  crest  and  the  dunes  were  reduced  In  height  (Annales 
des  ponts  et  chaussees,  5th  ser.,  vol.  1,  pp.  270-272, 1871). 


trough,  but  at  the  crest  they  have  an  up  and  down  motion 
as  well  as  a  forward  motion.  On  the  crest  of  the  larger 
waves  their  forward  motion  is  small  compared  with  their 
vertical  motion.  *  *  * 

There  is  a  sand  movement  by  rotation  or  whirls  that 
aids  transportation.  These  whirls  have  been  observed 
during  dune  motion  only  for  smaller  sizes  of  sand .  They 
are  of  short  duration,  lasting  usually  less  than  one  minute, 
but  in  this  time  one  of  these  may  scour  a  hole  1  to  3  inches 
deep  and  4  to  10  inches  in  length.  They  usually  start 
near  the  side  of  the  trough,  the  axis  inclining  downstream 
and  toward  the  center,  making  an  angle  of  30°  to  60°  with 
the  side  and  a  small  angle  at  the  bottom.  These  whirls 
are  3  to  5  inches  in  diameter  and  the  sand  grains  are  thrown 
violently  up  as  well  as  downstream  by  them.  This  move- 
ment aids  transportation  by  its  lifting  action,  some  of  the 
grains  being  carried  in  suspension  for  a  short  distance  by  it. 

The  change  in  the  appearance  of  a  loaded  stream  as  the 
load  is  increased,  the  discharge  remaining  constant,  is 
very  striking.  For  no  load  the  water  surface  is  even  and 
smooth.  As  fine  sand  is  fed  into  the  water  at  a  slow  rate, 
small  sand  dunes  will  form  on  the  bottom  and  many  little 
waves  will  form  on  the  surface.  As  the  rate  of  feeding  is 
increased,  the  slope  and  velocity  increasing,  these  waves 
become  larger  and  fewer  and  have  the  shape  of  an  inverted 
canoe.  These  canoes  are  side  by  side,  the  number 
depending  on  the  trough  width  and  size  of  waves.  When 
the  width  was  1.0  foot  two  sets  formed  side  by  side;  when 


THE   OBSERVATIONS. 


33 


the  width  was  1.32  feet  three  sets  formed.  As  the  critical 
velocity  at  which  dune  motion  ceases  is  approached  these 
waves  begin  todisappear,  and  when  thisvelocity  is  reached 
the  water  surface  is  waveless.  This  waveless  condition 
continues  as  the  rate  of  feed  increases  until  sand  motion 
in  antidunes  begins,  when  large  waves,  the  width  of  the 
trough  and  corresponding  in  length  to  the  sand  waves 
beneath  them,  are  formed  as  illustrated  in  figure  11. 

In  order  to  show  the  magnitude  of  these  surface  waves, 
wave  traces  have  been  drawn.  Some  of  these  are  given 
in  figure  12.  A  sheet  of  galvanized  iron  4  feet  long  and  1 
foot  high  was  divided  into  inch  squares  by  lines.  This 
plate  was  moistened  and  covered  with  fine  dust.  It  was 
held  vertical  at  a  given  place  over  the  experiment  trough 
and  on  signal  was  dropped  into  the  trough  and  taken  out 
again  as  quickly  as  possible.  The  dust  was  removed  from 
that  part  of  the  plate  in  the  water,  leaving  a  well-defined 
outline  of  the  wave.  This  wave  trace  was  quickly  sketched 
on  paper  by  means  of  the  lines  marking  the  squares.  Trace 
A,  figure  12,  is  for  zero  feed ;  B  is  for  a  very  small  feed ;  C  is 
for  a  larger  feed,  the  surface  being  covered  with  the  canoe- 
shaped  waves ;  D  shows  one  of  the  larger  waves  associated 
with  autidunes. 


The  slopes  at  which  the  phases  of  traction 
change  are  lower  for  large  streams  than  for 
small,  and  lower  for  fine  debris  than  for  coarse. 
The  phase  with  smooth  bed — which  may  con- 
veniently be  called  the  smooth  phase — covers 
a  greater  range  of  conditions  with  mixed 
debris  than,  with  assorted. 

The  processes  associated  with  dunes  and 
antidunes  were  briefly  studied  in  the  glazed 
trough  and  with  the  moving  field.  Trans- 
portation by  saltation  follows  the  entire  pro- 
file of  the  antidunes  but  traverses  the  dunes 
only  from  A  to  C  of  figure  10.  The  velocity 
of  saltatory  grains  is  greatest  where  erosion 
takes  place,  namely,  along  the  upstream  slopes 
of  dunes  and  the  downstream  slopes  of  anti- 
dunes,  and  it  may  reasonably  be  inferred  that 
the  water  velocities  are  greatest  in  those 
places.  The  eye  detected  no  difference  in  water 


21  2Z  23 

FIGURE  12.— Profiles  of  water  surface,  automatically  recorded,  showing  undulations  associated  with  the  antidune  mode  of  traction.    Numbers 

show  distance,  in  feet,  from  the  head  of  experiment  trough. 


depth  over  the  two  slopes  of  the  antidune,  and 
if  the  depth  is  the  same  so  also  is  the  mean 
velocity;  but  the  ratio  of  bed  velocity  to  mean 
velocity  is  known  to  vary  with  conditions. 

The  cause  of  the  changes  in  process  has  not 
been  adequately  investigated,  but  a  few  sug- 
gestions may  be  made.  To  assist  in  a  search 
for  controlling  conditions,  the  factors  con- 
nected with  the  two  critical  points — the  change 
from  dune  phase  to  smooth  and  the  change 
from  smooth  to  antidune — were  tabulated 
from  the  experiments  with  sand  of  a  single 
grade  (C) ;  the  positions  of  the  critical  points 


being  estimated  by  Mr.  Murphy  at  a  time  when 
the  details  of  the  experiments  were  freshly  in 
mind.  In  Table  3  w  is  the  width  of  trough,  in 
feet ;  Q  the  discharge,  in  cubic  feet  per  second ; 
8  the  per  cent  of  slope;  d  the  depth  of  water  in 
feet;  Vm  the  mean  velocity,  in  feet  per  second; 
L  the  load,  in  grams  per  second;  and  Ll  the 
load  per  unit  width.  The  data  in  this  table 
are  taken  from  a  preliminary  reduction  of  the 
observations  and  are  less  accurate  than  the 
results  of  the  final  adjustment,  which  appear 
in  Table  12  (p.  75).  They  suffice,  however, 
for  the  present  purpose. 


TABLK  3. — Data  connected  with  changes  in  mode  of  transportation. 


First  critical  point. 

Second  critical  point. 

w 

S 

d 

Ym 

L 

Li 

S 

A 

vm 

L 

£i 

0.66 

0.093 

1.00 

0.076 

1.86 

13 

20 

1.86 

0.058 

2.44 

44 

67 

.66 

.182 

.93 

.115 

2.41 

28 

46 

1.72 

.092 

3.01 

88 

134 

.66 

.363 

.82 

.190 

2.91 

48 

73 

1.51 

.165 

3.37 

145 

221 

.66 

.545 

.74 

.272 

3.05 

70 

106 

1.34 

.209 

3.97 

184 

281 

.00 

.182 

1.15 

.082 

1.76 

41 

41 

2.02 

.056 

3.25 

111 

111 

.00 

.363 

.89 

.134 

2.71 

70 

70 

1.70 

.109 

3.33 

201 

201 

.00 

.545 

.81 

.180 

3.03 

95 

95 

1.51 

.142 

3.75 

244 

244 

.00 

.734 

.73 

.227 

3.24 

113 

113 

1.34 

.194 

3.78 

301 

301 

.32 

.182 

1.20 

.068 

2.03 

33 

25 

2.15 

.051 

2.70 

121 

92 

1.32 

.363 

1.00 

.107 

2.57 

76 

57 

1.76 

.099 

2.77 

210 

159 

1.32 

.545 

.87 

.145 

2.84 

106 

80 

1.53 

.111 

3.71 

251 

190 

1.32 

.734 

.75 

.178 

3.13 

140 

106 

1.33 

.149 

3.73 

293 

222 

1.96 

.363 

1.04 

.081 

2.29 

67 

34 

1.91 

.062 

2.99 

219 

112 

1.96 

.734 

.84 

.133 

2.82 

138 

71 

1.48 

.108 

3.47 

354 

181 

1.96 

1.119 

.74 

.182 

3.14 

191 

98 

1.22 

.150 

3.81 

418 

213 

20021°— No.  80— 14- 


34 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


The  factors  were  then  plotted  in  various 
combinations  on  logarithmic  section  paper,  and 
certain  approximate  numerical  relations  were 
thus  discovered.  The  first  critical  point  is 
reached  when  d  =  0.016  Fm2'3,  or  when  d  = 
0.0045  i,°'85.  The  second  critical  point  is 
reached  when  d  =  0.004  Fm3'3,  or  when  <Z  = 
0.0003  L,1'15.  The  coefficients  and  exponents 
are  not  well  defined  by  the  data,  but  the 
general  indications  are  (1)  that  each  change  in 
phase  occurs  when  the  depth  of  water  bears 
a  certain  numerical  relation  to  a  power  of  the 
mean  velocity  near  the  cube,  and  (2)  that  the 
changes  occur  when  the  depth  bears  a  certain 
numerical  relation  to  the  amount  of  load 
carried  in  each  unit  of  width  of  current. 

At  the  bottom  the  stream  is  limited  and 
restricted  by  the  bed  of  debris;  at  the  top  by 
the  water  surface.  To  the  space  between  these 
bounds,  a  space  measured  by  the  depth,  the 
eddies  or  convolutions  of  the  current  are  con- 
fined. Within  the  range  of  conditions  covered 
by  the  experiments  the  normal  mode  of  flow 
involves  sinuosity  of  the  filaments  of  current, 
and  the  tendency  toward  diversity  of  internal 
movement  is  strong  in  proportion  as  the  veloc- 
ity is  high.  A  particular  relation  between 
depth  and  velocity  corresponds  to  a  sort  of 
equilibrium  between  the  factors  of  turbulence 
and  restraint,  in  accordance  with  which  the 
sinuosity  of  the  lines  of  flow  is  reduced  to  a 
minimum  and  the  water  surface  and  channel 
bed  are  approximately  plane.  This  gives  the 
smooth  phase  of  traction.  When  the  depth  is 
increased  without  increase  of  velocity,  the 
reduction  of  restraint  permits  the  develop- 
ment of  internal  diversity,  and  this  carries  with 
it  diversity  of  the  plastic  bed,  giving  the 
dune  phase  of  traction.  When  the  velocity 
is  increased  without  increase  of  depth,  the 
restraint  is  overpowered,  and  a  diversified  but 
systematic  arrangement  of  flow  lines  develops, 
which  carries  with  it  systematic  diversity  of 
both  water  surface  and  channel  bed  and  gives 
the  antidune  phase. 

There  will  be  occasion  to  speak  of  these 
relations  in  another  connection  in  Chapter 
XIV. 

It  may  be  noted,  as  a  possible  contribution 
toward  an  explanatory  analysis,  that  gravity 
opposes  the  current  on  the  upstream  side  of 
the  antidune  (fig.  11)  and  assists  the  current  on 


the  downstream  side.  It  is  where  gravity  ac- 
celerates that  load  is  increased  by  erosion,  and 
where  gravity  retards  that  load  is  reduced  by 
deposition.  In  the  case  of  the  dune,  however, 
erosion  occurs  where  gravity  is  a  retarding 
force.  But  here  the  descending  current,  which 
is  accelerated  by  gravity  and  which  is  observed 
to  have  gained  speed,  is  free  from  the  bed  and 
bears  no  load  (fig.  10).  In  order  to  transport 
when  it  resumes  contact  with  the  bed  it  must 
take  debris  from  the  bed,  and  by  so  taking  it 

erodes. 

UNITS. 

The  system  of  units  to  which  the  laboratory 
data  have  been  reduced  and  which  will  bo  em- 
ployed in  their  discussion  is  hybrid  in  that  it 
includes  the  foot  and  the  gram.  The  foot  is 
made  the  fundamental  unit  for  length,  area, 
and  volume  because  it  is  the  unit  employed  gen- 
erally by  English-speaking  engineers.  The 
gram  is  made  the  unit  of  mass,  primarily  be- 
cause it  is  of  convenient  magnitude,  but  also 
because  of  the  manifest  advantage  of  intro- 
ducing the  metric  system  wherever  no  practical 
difficulties  interfere.  It  happens  that  the  two 
measures  which  are  given  in  grams  are  of  cate- 
gories unfamiliar  alike  to  the  engineer  and  the 
general  reader,  so  that  the  gram  unit  encounters 
no  conflicting  habit  of  thought.  One  measure 
is  the  mass  of  a  grain  of  sand  or  a  pebble,  the 
other  the  mass  of  the  debris  carried  by  a  stream 
hi  a  second. 

The  unit  of  tune,  for  the  indication  of  rates, 
is  one  second.  Velocity  is  given  hi  distance  per 
second,  ft./sec. ;  discharge  in  volume  per  second, 
f t.3/sec. ;  and  load  in  mass  per  second,  gm./sec. 

In  hydraulic  and  hydrodynamic  treatises 
slope  of  streams  is  measured  by  the  quotient  of 
fall  by  distance,  or  the  tangent  of  its  angle,  and 
the  unit  slope  is  taken  as  45°.  For  practical 
purposes  this  unit  is  inconvenient  because  it 
transcends  experience,  and  engineers  commonly 
avoid  it  by  speaking  of  slope  in  percentage  or 
in  fractions  of  1  per  cent,  thus  making  1  per 
cent  the  actual  unit.  For  most  purposes  I 
find  the  smaller  unit  most  convenient,  but  I 
have  occasional  use  for  the  larger  unit  and  shall 
accordingly  use  both.  To  avoid  confusion  the 
symbol  8  will  be  used  with  the  smaller  unit, 
and  where  discrimination  is  important  it  will 
be  called  per  cent  slope,  while  the  symbol  s  will 
'be  used  with  the  larger  unit.  S  — 100  s. 


THE   OBSEBVATIONS. 


35 


TERMS. 

Load. — The  quantity  of  debris  transported 
by  a  stream  through  any  cross  section  iu  a  unit 
of  time  is  its  load  at  that  section.  A  part  is 
carried  in  suspension  and  a  part  by  traction, 
but  as  we  are  here  concerned  with  traction 
only,  the  fractional  load  is  to  be  understood 
when  the  word  is  used  without  specification. 
Load  is  measured  in  grams  per  second,  gm./sec. 
For  certain  engineering  purposes  it  is  desirable 
to  consider  load  as  volume,  not  as  the  sum  of 
the  volumes  of  individual  grains,  but  as  the 
gross  space  occupied  by  the  debris  as  a  natural 
deposit.  For  the  debris  used  in  the  experi- 
ments in  stream  traction,  with  the  mixture 
of  sizes  ordinarily  found  in  a  river  deposit,  the 
weight  of  1  cubic  foot  is  about  50,000  grams,  or 
110  pounds.  The  symbol  for  load  is  L. 

Capacity. — The  maximum  load  a  stream 
can  carry  is  its  capacity.  It  is  measured  in 
grams  per  second,  gm./sec.  As  the  work  of 
the  laboratory  was  largely  to  determine 
capacity  by  measuring  maximum  load,  the 
two  terms  are  to  a  large  extent  interchangeable 
in  the  discussion  of  laboratory  data,  but  the 
distinction  is  nevertheless  important.  -The 
symbol  for  capacity  is  0. 

Capacity  is  a  function  of  various  conditions, 
such  as  slope  and  discharge,  and  the  chief 
purpose  of  the  laboratory  investigation  was  to 
discover  the  relations  of  capacity  to  conditions. 
When  a  fully  loaded  stream  undergoes  some 
change  of  condition  affecting  its  capacity, 
it  becomes  thereby  overloaded  or  underloaded. 
If  overloaded,  it  drops  part  of  its  load,  making 
a  deposit.  If  underloaded,  it  takes  on  more 
load,  thereby  eroding  its  bed.  Through  these 
reactions  the  profiles  of  stream  beds  are 
adjusted,  so  far  as  stream  beds  are  composed  of 
debris.  If  the  bed  is  of  rock  in  place,  the  under- 
loaded stream  can  not  obtain  its  complement 
of  debris,  but  nevertheless  it  attacks  the  bed. 
By  dragging  debris  over  the  rock  it  files  or 
corrades  the  bed  of  its  channel.  It  is  a  general 
fact  that  the  loads  of  streams  flowing  on  bed- 
rock are  less  than  their  capacities. 

Competence. — Under  certain  combinations  of 
controlling  factors  capacity  is  zero  or  negative. 
If  then  some  one  factor  be  changed  just  enough 
to  render  capacity  positive,  that  factor  in  its 
new  condition  is  said  to  be  competent,  or  else 


to  be  a  measure  of  the  stream's  competence. 
For  example,  a  stream  at  its  low  stage  can  not 
move  the  debris  on  its  bed;  with  increase  of 
discharge  a  velocity  is  acquired  such  that 
traction  begins;  and  the  discharge  is  then  said 
to  be  competent.  A  stream  flowing  over 
a  too  gentle  slope  has  no  capacity,  but  coming 
to  a  steeper  slope  it  is  just  able  to  move  debris; 
the  steeper  slope  is  said  to  be  competent.  A 
current  flowing  over  debris  of  various  sizes 
transports  the  finer  but  can  not  move  the 
coarser;  the  fineness  of  the  debris  it  can  barely 
move  is  the  measure  of  its  competence. 

Discharge. — The  quantity  of  water  passing 
through  any  cross  section  of  a  stream  in  a  unit 
of  time  is  the  discharge  of  the  stream  at  that 
point.  It  is  measured  in  cubic  feet  per  second, 
ft.3/sec.  The  symbol  is  Q. 

Slope. — The  inclination  of  the  water  surface 
in  the  direction  of  flow  is  known  as  the  slope  of 
the  stream.  It  is  the  ratio  which  fall,  or  loss 
of  head,  bears  to  distance  in  the  direction  of 
flow.  Per  cent  slope,  as  explained  on  page  34, 
is  numerically  100  tunes  as  great, .being  the 
fall  in  a  distance  of  100  units.  The  terms 
slope  and  per  cent  slope  are  also  applied  in 
this  report  to  the  inclination  of  the  bed  of  the 
channel. 

Size  of  debris. — The  relative  magnitude  of 
the  debris  particles  making  up  the  load  may  be 
considered  from  two  opposed  viewpoints. 
Thus  we  may  say  that  the  load  varies  inversely 
with  the  coarseness  of  the  debris,  or  that  it 
varies  directly  with  the  fineness.  The  second 
viewpoint  is  here  preferred  because  it  conduces 
to  symmetry  in  formulation.  Two  very  dif- 
ferent measures  of  fineness  have  been  con- 
sidered, and  on  page  21  the  material  of  the 
laboratory  is  listed  under  both.  One  defines 
fineness  by  the  number  of  particles  which, 
placed  side  by  side,  occupy  the  linear  space  of 
1  foot.  The  other  defines  it  by  the  number  of 
particles  required  to  fill  the  space  of  1  cubic 
foot.  Linear  fineness  is  the  more  readily 
conceived,  because  it  appeals  to  vision.  Bulk 
fineness  is  the  more  easily  determined.  The 
symbol  of  linear  fineness  is  F,  of -bulk  fine- 
ness Ft. 

Form  ratio. — Of  the  variable  factors  which 
in  combination  produce  the  multifarious  chan- 
nel forms  of  natural  streams,  the  laboratory 
dealt  extensively  with  but  a  single  one,  the 


36 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


relation  of  depth  to  width.  The  relation  is  a 
simple  ratio,  and  either  of  the  two  terms 
might  be  made  the  divisor.  The  width  has 
been  chosen  because,  as  in  the  case  of  fineness 
its  selection  conduces  to  symmetry  in  formula- 

tion.     The  ratio  will  be  called 

form  ratio.     Its  symbol  is  R. 

Duty  and  efficiency. — Capacity  varies  with 
discharge,  but  is  not  proportional  to  it.  The 
load  which  may  be  borne  by  a  unit  of  dis- 
charge varies  with  the  discharge  and  also  with 
the  other  conditions.  It  is  the  capacity  per 
unit  discharge,  or  the  quotient  of  capacity  by 
discharge,  and  will  be  called  the  duty.  The 
symbol  is  U. 

Capacity  varies  also  with  slope  but  is  not 
proportional  to  it.  As  the  product  of  discharge 
by  slope  (by  the  acceleration  of  gravity) 
measures  the  stream's  potential  energy  per 
unit  time  per  unit  distance,  capacity  also  varies 
with  the  energy  but  is  not  proportional  to  it. 
The  load  which  may  be  borne  by  a  unit  of  dis- 
charge on  a  unit  slope  varies  with  all  the  con- 
ditions of  transportation.  It  is  the  capacity 
per  unit  discharge  and  unit  slope,  or  the  quo- 
tient of  capacity  by  the  product  of  discharge 
and  slope,  and  will  be  called  the  efficiency. 
It  is  a  measure  of  the  stream's  potential  work 
of  transportation  in  relation  to  its  potential 
energy.  Its  symbol  is  E. 

The  primary  definition  of  efficiency  in  me- 
chanics is  the  ratio  of  work  done  to  energy  ex- 
pended; it  implies  that  the  work  may  be  ex- 
pressed in  the  same  unit  as  energy.  The  ratio 
is  always  less  than  unity.  But  there  is  an 
important  secondary  use  of  the  term,  applied 
to  cases  in  which  the  result  accomplished  can 
not  be  expressed  in  terms  of  energy.  In  such 
cases  the  ratio  may  have  any  magnitude,  as  it 
arises  from  the  comparison  of  incongruous 
quantities.  It  does  not  measure  economy  of 
energy  but  relative  accomplishment  in  respect 
to  any  condition  selected  for  comparison.  As 
capacity  for  transportation  is  not  statable  in 
units  of  energy,  the  use  of  the  term  efficiency 
in  this  connection  falls  under  the  second  defini- 
tion. 

Symbols. — An  index  of  symbols,  with  brief 
definitions  and  references  to  pages  for  fuller 
definition,  may  be  found  on  page  13. 


TABLE  OF  OBSERVATIONS  ON  STREAM 
TRACTION. 

The  observations  on  stream  traction  &re 
presented  in  Table  4.  As  the  original  notes 
are  voluminous,  certain  combinations  and  re- 
ductions were  made  before  tabulation.  The 
reduction  of  the  slope  observations  involved 
discrimination,  and  the  mode  of  reduction  is 
therefore  described  below.  Each  horizontal 
line  of  the  table  contains  the  record  of  a  single 
experiment. 

The  observations  are  arranged  according  to 
(1)  fineness  of  dfibris,  (2)  width  of  channel, 
(3)  discharge,  (4)  load  and  slope.  The  cate- 
gories of  fineness  begin  with  single  sizes, 
taken  in  order  from  fine  to  coarse,  and  follow 
with  mixtures;  they  constitute  subtables,  each 
designated  by  a  letter  in  parentheses.  The 
arrangement  by  load  and  slope  is  approximate 
only.  In  a  general  way  the  sequences  of  load 
and  slope  are  parallel,  each  increasing  as  the 
other  increases,  but  the  data  are  not  per- 
fectly harmonious,  and  where  the  two  se- 
quences differ  the  arrangement  is  somewhat 
irregular. 

The  first  column,  for  all  divisions  of  the 
table  except  (J),  gives  width  of  trough;  the 
second,  discharge.  The  third,  fourth,  and 
fifth  pertain  to  load  and  give,  respectively, 
the  load  as  measured  by  debris  fed  at  the 
head  of  the  trough,  the  load  as  measured  by 
debris  caught  at  the  outfall  end,  and  the 
period,  in  minutes,  during  which  debris  was 
collected  at  the  outfall  end.  The  precision  of 
measurement  is  probably  somewhat  higher 
where  the  period  is  relatively  long. 

The  next  three  columns  pertain  to  slope. 
The  sixth  contains  the  slope,  in  per  cent,  of 
the  water  surface;  the  seventh,  the  slope  of 
the  bed  of  debris  as  shaped  by  the  current; 
and  the  eighth,  the  extreme  distance  between 
points  at  which  were  made  observations  used 
in  computing  the  slope  or  slopes.     Nearly  all 
the  experiments  for  which  the  recorded  dis- 
tance is    16   feet   or  less  were   made   in   the 
shorter  trough,  of  which  the  gross  length  was 
31.5   feet,    the   distance   between   the   debris- 
feeding  apparatus  and  the  debris-arresting  ap- 
paratus being  24.5  feet.     The  experiments  for 
which  the  recorded  distance  exceeds   16  feet 
were  made  in  the  longer  trough,  the  distance 


THE   OBSERVATIONS. 


37 


from  the  feeding  station  to  the  collecting  sta- 
tion being  at  least  16  feet  longer  than  the 
space  covered  by  the  observations  used  in  the 
computation. 

The  ninth  column  shows  the  depth  at  a 
single  point  as  measured  by  the  gage  (see 
p.  21);  the  tenth  gives  the  mean  depth  as 
estimated  from  the  records  of  water  profile  and 
bed  profile  (see  p.  25). 

In  the  eleventh  column,  headed  "Character 
of  bed,"  a  generalization  is  given  from  notes 
on  the  mode  of  transportation,  the  condition 
of  the  water  surface  during  the  run,  and  the 
condition  of  the  bed  of  debris  after  the  with- 
drawal of  the  water.  The  original  notes  are 
somewhat  varied  in  scope  and  nomenclature, 
and  it  seemed  best  to  make  the  tabulated 
record  more  simple.  The  words  dunes,  smooth, 
and  antidunts  denote  the  three  modes  of  trac- 
tion described  on  pages  30-33.  The  word 
transition  is  used  where  the  mode  of  traction 
was  intermediate  between  one  of  the  rhythmic 
modes  and  the  smooth  mode,  and  also  where 
different  modes  of  traction  obtained  in  differ- 
ent parts  of  the  trough. 

In  the  final  column  the  word  free  indicates 
that  the  contractor  (p.  25)  was  not  used,  but 
the  experiment  trough  retained  its  full  width 
to  the  end. 

The  reduction  of  the  slope  observations  was 
preceded  by  a  careful  study  of  them  with  refer- 
ence to  their  systematic  and  accidental  errors. 
As  a  result  of  that  study  certain  criteria  of 
exclusion  were  adopted,  by  means  of  which,  it 
is  thought,  the  influence  of  systematic  errors 
was  materially  reduced.  The  criteria  were 
applied  through  an  inspection  of  the  plotted 
profiles  of  water  surface  and  de'bris  bed.  The 
observations  excluded  were  those  believed  to 
be  much  affected  either  by  the  peculiar  con- 
ditions near  the  head  of  the  trough  or  by  the 


peculiar  conditions  near  the  outfall  end,  so 
that  the  retained  observations  constituted  a 
continuous  series  covering  the  middle. 

The  observations  to  be  reduced  were  as- 
cribed equal  weights.  They  consisted  of  a 
series  of  level  readings  (ha,  7ilt  Ti2  .  .  .  7in^, 
hn),  each  giving  the  vertical  distance  of  a  point 
of  the  de'bris  surface,  or  of  the  water  surface, 
below  a  horizontal  plane  of  reference,  together 
with  the  horizontal  distances  of  the  same 
points  from  an  initial  point  in  the  axis  of 
the  trough.  The  observed  points  being  at 
equal  intervals  along  the  trough,  and  the  zero 
of  distances  being  made  coincident  with  the 
first  of  the  observed  points,  the  horizontal  dis- 
tances may  be  represented  by  0,  I,  2l,  .  .  . 
(n—l)l.  The  following  formula1  was  used  to 
compute  the  slope,  the  number  of  terms  in 

7? 

numerator  or  denominator  being  x  when  n 
was  an  even  number,  and  — ^ —  when  n  was  odd. 


„    100 (hn-h0)n+(hn_-hl)(n-2)+(hn^-h1)(n-4)+  .  .  .  ,, , 
I    '  n2+(n-2)2+(n-4)2+  .  .  . 


The  numerical  operations  were  simple. 

In  division  (J)  of  the  table  the  arrangement 
is  somewhat  different,  and  a  column  is  added 
at  the  left.  The  division  records  experiments 
made  with  debris  prepared  by  mixing,  in  defi- 
nite proportions,  two  or  more  of  the  grades  of 
debris  to  which  the  preceding  divisions  pertain. 
The  notation  adopted  to  designate  these  mix- 
tures is  analogous  to  that  for  chemical  com- 
pounds, subscript  figures  being  used  to  indicate 
approximate  proportions.  This  notation,  to- 
gether with  the  more  precise  indication  of  the 
proportions,  is  given  in  the  left-hand  column. 

i  EssentiaUy  a  least-squares  formula,  although  developed  from  appa- 
rently independent  considerations. 


38  TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 

TABLE  4  (A). — -Observations  on  load,  slope,  and  depth,  with  debris  having  25,tOO  particles  to  the  gram,  or  grade  (A). 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
0.66 

Fl.'lsec. 
0.093 

Gm.lsec. 
27 
67 

dm.  /sec. 
24 
74 

Minutes. 
5 
5 

Per  cent. 
1.12 

Per  cent. 
1.22 
2.14 

Feet. 
16 
16 

Feel. 
0.065 

Feet. 
0.083 

Contracted. 
Do. 

.182 

19 
93 

18 
96 

10 

4 

.69 
1.59 

16 
14 

.12K 
.095 

Smooth     .  . 

Contracted. 
Do. 

Antidunes  

.545 

59 
147 

58 
157 

5 
3 

.56 

.55 
1.07 

16 
16 

.272 

.284 

Smooth  ....          

Contracted. 
Do. 

1.00 

.182 

10.5 
151 

9.4 
155 

10 
3 

.58 

.53 
2.04 

16 
16 

.118 

.136 

Contracted. 
Do. 

Antidunes  

.363 

20 
240 

19 
245 

8 
3 

.42 
1.64 

14 
16 

.175 

Transition...  . 

Contracted. 
Do. 

Antidunes  

.734 

52 
317 

51 
317 

4 
3 

.44 
1.19 

14 
16 

.264 

Contracted. 
Do. 

Antidunes. 

1.32 

.182 

12 
17 
18 
23 
37 
50 
75 
73 
129 
120 

12 
18 
17 
21 
38 
51 
68 
73 
120 
135 

10 
10 
10 
8 
6 
5 
5 
5 
5 
3 

.65 
.83 

.63 
.75 
.73 
.79 
.99 
1.10 
1.37 
1.41 
1.79 
1.86 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.091 

.084 
.075 
.080 
.063 
.058 
.061 
.060 

.106 
.091 

Dunes 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do 

.do 

.80 

.083 
.087 
.079 

do 

1.11 

Antidunes  . 

Antidunes..   .. 

do  

.363 

7.3 
42 
45 
44 
65 
69 
125 
121 

8.9 
38 
41 
43 
74 
76 
133 
140 
177 
202 
240 

8 
6 
5 
5 
4 
5 
4 
5 
4 
4 
4 

.37 

.60 

.27 
.51 
.58 
.67 
.84 
.90 
.18 
.14 
.43 
.55 
.65 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.191 
.124 
.115 
.112 
.112 
.130 
.115 
.106 
.089 
.097 

.  l«f> 
.131 

Dunes  

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

Transition  

.77 

Transition  

do... 

..     .do 

214 

r  Antidunes].  .  . 

.do. 

.734 

38 
82 
86 
127 
151 
191 
176 

38 
84 
89 
156 
154 
192 
215 
240 
327 
365 

5 
4 
4 
4 
3 
3 
3 
3 
3 
3 

.35 
.53 

.35 
.48 
.51 
.73 
.83 
.81 
.97 
.95 
1.13 
1.24 

10 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.267 
.207 

Dunes  

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

.205 
.198 
.190 

Smooth.  . 

.do 

Antidunes. 

do 

[Antidunes] 

-do... 

do 

[Antidunes] 

1.% 

.363 

8.1 
20 
19 
33 
58 
81 
•    128 
143 
128 
192 
215 
227 

8.7 
20 
21 
32 
66 
92 
127 
128 
144 
179 
193 
218 
264 
283 

6 
6 
6 
5 
5 
5 
5 
5 
4 
4 
4 
4 
3 
3 

.50 

.36 
.41 
.55 
.60 
.78 
.94 
1.10 
1.17 
1.18 
1.38 
1.50 
1.59 
1.73 
1.77 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.179 
.118 
.118 
.103 
.090 
.082 
.073 
.077 
.075 
.072 
.076 
.067 
.069 
.070 

.187 

Dunes.. 

Contracted  . 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

.69 
.67 

.78 
.97 

.129 
.107 
.097 
.084 

Antidunes.  .  . 

.do 

...do... 

do 

[Antidunes] 

do 

-do.   . 

.734 

7.7 
32 
74 
70 
105 
148 

8.3 
33 
88 

80 
105 
163 
167 
179 
231 
279 

10 
6 
4 
5 
5 
4 
4 
3 
3 
3 

.18 

.18 
.36 
.45 
.49 
.56 
.79 
.75 
.94 
.98 
1.01 

16 
12 
16 
16 
16 
16 
16 
16 
16 
16 

.283 
.180 
.156 
.156 
.137 
.142 
.135 
.129 
.135 
.130 

.293 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

.58 
.79 

.152 
.146 

Smooth... 

do  

Transition  .   .   . 

214 
221 

[  Antidunes]  

1.119 

94 
99 
186 
169 
180 
198 
221 

91 
102 
130 
188 
221 
229 
258 
291 
341 
346 

4 

4 
4 
3 
3 
3 
4 
3 
3 
3 

.30 

.45 
.39 
.57 
.59 
.63 
.60 
.81 
.79 
.95 
1.02 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.220 
.207 
.194 
.196 
.182 
.187 
.165 
.179 

.214 
""."261" 

Smooth  
[Smooth]  

Smooth. 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

.33 

do 

do 

Transition 

...do... 

.176 

do 

THE   OBSERVATIONS.  39 

TABLE  4  (B). — Observations  on  load,  slope,  and  depth,  inth  debris  having  15,400  particles  to  the  gram,  or  grade  (B). 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed  . 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

lied. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
0.23 

Ft.'/iec. 
0.093 

Om./sec. 

Gm.fsec. 
4.0 
6.7 
8.1 
20 
33 

Minutes. 
9 

7 

7 

Per  cent. 

Per  cent. 
0.85 
.94 
1.01 
1.49 
2.12 

Feet. 
16 
16 
16 
16 
16 

Feet. 
0.192 
.184 
.174 
.146 

Feet. 

Transition 

Free. 
Do. 
Do. 
Do. 
Do. 

do 

Smooth 

do 

.182 

4.0 
18 
31 
32 

10 
6 
5 

7 

.73 
1.12 
1.55 
1.55 

16 
16 
16 
16 

.364 
.284 
.230 
.232 

Free. 
Do. 
Do. 
Do. 

Transition 

do 

.44 
.66 

.093 

5.3 
11 
18 
34 
53 
64 

6 

7 
4 
7 
5 

4 

.73 
.90 
1.18 
1.62 
2.31 
2.38 

16 
16 
16 
16 
16 
16 

.136 
.110 
.093 
.070 
.072 
.068 

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 

do 

.do  

.182 

8.5 
16 
27 

73 
76 

7 
5 
6 
8 
4 

.50 
.75 
.98 
1.66 
1.73 

16 
16 
16 
16 

16 

.194 
.177 
.146 

Transition  ...     . 

Free. 
Do. 
Do. 
Do. 
Do. 

Smooth  

Transition  . 

.139 

.do  

.093 

5.1 
9.2 
17 
15 
22 
28 
34 
41 
53 
57 
93 

8 
6 
7 
9 
5 
5 
5 
6 
5 
5 

.75 
.84 
.23 
.32 
.41 
.47 
.63 
2.01 
2.09 
2.17 
2.96 

16 
16 
16 
12 
16 
16 
16 
16 
16 
16 
16 

.089 
.080 
.060 
.059 
.058 
.050 
.056 
.054 
.049 
.058 
.037 

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do 

Transition 

do... 

do... 

do 

do  



Antidunes  
do... 

do  

.182 

4.8 
3.5 
14 
15 
23 
27 
24 
42 
47 
41 
51 

4.2 
6.0 
12 
16 
20 
21 
19 
42 
39 
44 
47 
55 
55 
55 
67 
72 
88 
105 
112 
119 
126 
139 
159 
169 
213 
236 
254 
268 
377 

8 
8 
7 
6 
8 
4 
8 
7 
6 
9 
8 
5 
8 
5 
7 
7 
4 
5 
6 
6 
4 
5 
5 
5 
4 
3 
3 
4 
3 

0.32 
.37 

.35 
.36 
.69 
.68 
.66 
.72 
.81 
.98 
1.06 
1.18 
1.17 
1.32 
1.35 
1.36 
1.51 
1.54 
1.51 
2.00 
1.88 
2.05 
2.24 
2.19 
2.46 
2.79 
3.14 
3.06 
2.91 
2.93 
4.03 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.175 
.175 

0.183 
.183 

Contracted. 
Do. 
Do. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do.   ... 

.137 
.113 
.110 
.107 
.113 
.101 

.112 
.107 

Transition  

55 

.103 

.do  

81 

.do  

.091 
.082 
.083 
.077 

Antidunes  .'  

.do  

166 

Antidunes  

.115 
.0% 
.082 

[Antidunes]  

...do  

do... 

.do  

do  

.do  

...do  

.do  

.363 

29 
44 
52 
58 
73 

29 
40 
39 
49 
55 
77 
91 
104 
148 
152 
164 
167 
186 
226 

5 
8 
6 
6 
5 
4 
5 
5 
4 
5 
4 
5 
3 
4 

.51 
.68 
.69 
.75 
.80 
.98 
.99 
1.14 
1.54 
1.43 
1.63 
1.66 
1.71 
1.81 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
14 
16 
16 

.216 
.212 
.205 
.202 
.188 
.178 
.189 
.191 
.177 
.155 

.221 

Contracted. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

...do  

do  

do  

Transition 

Antidunes  

..  ..do  

[Antidunes)  

...do  

do... 

.158 

...do  

do... 

..do  

.545 

5.8 
16 
35 
54 
68 
81 
104 
111 
132 
169 
242 
271 
304 

6 
9 
5 
8 
6 
6 
5 
5 
5 
5 
3 
3 
3 

.18 

.21 
.23 
.46 
.63 
.74 
.72 
.87 
.89 
1.00 
1.32 
1.81 
1.81 
1.79 

16 
14 
14 
16 
16 
14 
16 
16 
16 
16 
16 
16 
16 

.471 
.367 
.291 
.292 
.258 
.260 
.220 
.226 
.230 
.231 

.472 

Contracted. 
Do. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

16 
34 

64 
67 
79 
108 
105 

do  

.47 

.300 

do  

.do  

Transition  

Antidunes  

do  

[  Antidunes)  

do  

...do  

40  TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 

TABLE  4  (B). — Observations  on  load,  slope,  and  depth,  with  debris  having  13,400  particles  to  the  grim,  or  grade  (B)— Con. 


Load. 

Slope. 

Dep 

th. 

Width. 

charge. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 

Ft.'lsec. 

Om.leec. 
9  9 

Qm.jsec. 
11 

Minutes. 
5 

Per  cent. 

Per  cent. 
0  61 

Feet. 
16 

Feet. 
0  106 

Feet. 

Contracted. 

9  9 

13 

5 

.61 

16 

.102 

Do. 

28 

6 

.97 

16 

.073 

Transition  .  .  . 

Free. 

39 

5 

1  15 

16 

.081 

do                   

Do. 

51 

5 

1  29 

16 

073 

Do. 

98 

4 

1  87 

16 

.000 

Do. 

116 

4 

2.14 

16 

.067 

do  .. 

Do. 

157 

3 

2.43 

16 

.066 

..      do                

Do. 

182 

4 

2.63 

16 

.070 

[Antidunes]  

Do. 

363 

o 

o 

0  030 

044 

60 

.475 

Contracted. 

2  1 

1  3 

390 

18 

.16 

60 

.344 

.323 

[Dunes]        

Do. 

1  9 

1  6 

360 

17 

.16 

60 

.321 

.321 

do 

Do. 

6.4 

6  2 

60 

.28 

48 

do 

Do. 

6  8 

6  8 

44 

.27 

.31 

44 

.278 

.271 

do 

Do. 

10  5 

7.2 

5 

.23 

16 

.214 

.244 

do 

Do. 

9  6 

9  4 

71 

.33 

40 

.224 

do 

Do. 

8.7 

9.7 

6 

.29 

16 

.208 

...  do  

Do. 

9  3 

11 

62 

.31 

44 

.254 

Do. 

18 

21 

30 

.42 

.43 

40 

.188 

.193 

Do. 

25 

25 

15 

.43 

.49 

36 

.195 

.176 

Do. 

28 

28 

4 

.54 

16 

.164 

Do. 

37 

49 

g 

.63 

32 

Transition 

Do. 

46 

4 

.78 

14 

.154 

Free. 

55 

9 

.65 

12 

.150 

do                       .      .  .. 

Do. 

79 

4 

1.00 

16 

.120 

Antidunes  

Do. 

153 

4 

1.51 

16 

.107 

..  do.. 

Do. 

206 

4 

1.66 

16 

.129 

do 

Do. 

218 

3 

1.78 

16 

.112 

.    do     . 

Do. 

268 

3 

2.00 

16 

do 

Do. 

305 

3 

2.14 

16 

[Antidunes]  

Do. 

347 

3 

2.28 

16 

do 

Do. 

545 

107 

5 

.79 

12 

.194 

Smooth 

Free. 

117 

5 

.83 

16 

.167 

Transition  

Do. 

167 

4 

1.16 

16 

.157 

Do. 

199 

3 

1.31 

16 

.169 

do                             * 

Do. 

219 

3 

1.29 

16 

[Antidunes]                 .  .  .. 

Do. 

274 

3 

1.46 

16 

Do. 

306 

3 

1.63 

16 

.144 

...  do  .              

Do. 

315 

3 

1.65 

16 

Do. 

396 

<y 

1.84 

16 

do 

Do. 

504 

2 

2.10 

16 

do 

Do. 

734 

0 

0 

.012 

40 

.735 

.749 

Contracted. 

3  9 

4  7 

60 

.14 

.15 

40 

.497 

.509 

Do. 

4.7 

4.9 

90 

.14 

.18 

36 

.496 

.497 

Do. 

18 

17 

20 

.25 

.26 

28 

.332 

.334 

Do. 

37 

38 

10 

.36 

.38 

24 

.282 

.267 

Do. 

56 

52 

10 

.41 

.48 

28 

.257 

.256 

Do. 

54 

8 

.45 

13 

.257 

Transition  

Free. 

56 

59 

10 

.50 

28 

.257 

Contracted. 

60 

7 

.53 

16 

.262 

Free. 

62 

9 

.44 

16 

.257 

...  do 

Do. 

84 

80 

5 

.31 

.54 

28 

•    .250 

.241 

Contracted. 

102 

5 

.68 

16 

.237 

Transition 

Free. 

188 

5 

.92 

16 

.207 

Do. 

244 

3 

1.16 

16 

Do 

305 

3 

1.30 

16 

do 

Do. 

370 

2 

1.46 

16 

Do 

416 

2 

1.72 

16 

.    do 

Do. 

433 

2 

1.56 

16 

do 

Do. 

1  32 

182 

7  3 

5.2 

8 

.40 

.40 

16 

.129 

.126 

5.8 

5.3 

8 

.45 

16 

[Dunes]  

Do. 

14 

6 

.80 

16 

.081 

Free. 

17 

6 

.98 

16 

.061 

....do  

Do. 

34 

6 

1.12 

16 

.063 

Transition 

Do. 

27 

9 

1.18 

12 

.057 

do  

Do. 

43 

6 

1.22 

16 

.056 

Smooth 

Do. 

56 

6 

1.41 

16 

.053 

Do. 

68 

6 

1.66 

16 

.041 

do 

Do. 

81 

5 

1.71 

16 

.051 

do 

Do. 

80 

5 

1.77 

16 

.038 

.do                 

Do. 

93 

7 

1.88 

16 

.045 

Do. 

115 

4 

2.11 

16 

.058 

..  do. 

Do. 

127 

4 

2.33 

16 

do 

Do. 

142 

4 

2.32 

16 

.037 

[Antidunes]  

Do. 

152 

4 

2.46 

16 

.039 

do 

Do. 

363 

8.3 

5.2 

5 

.21 

.25 

16 

.227 

.220 

Dunes 

Contracted. 

8  7 

7.9 

9 

.42 

.26 

16 

.203 

.197 

Do. 

17 

17 

9 

.53 

.43 

16 

.153 

.163 

..  do 

Do. 

19 

16 

9 

.64 

.43 

16 

.147 

.163 

do 

Do. 

22 

6 

.47 

14 

.131 

Free. 

22 

6 

.66 

16 

.128 

Do. 

25 

4 

.69 

14 

.124 

Dunes  .   . 

Do. 

38 

6 

.82 

12 

.109 

do 

Do. 

89 

1.07 

16 

.098 

Do. 

104 

6 

1.20 

16 

.099 

Do. 

138 

5 

1.28 

16 

.080 

Antidunes.  .  . 

Do. 

131 

5 

1.33 

16 

.072 

.      do 

Do. 

134 

5 

1.37 

16 

.098 

[Antidunes].  .  . 

Do. 

THE   OBSERVATIONS.  41 

TABLE  4  (B) — Observations  on  load,  slope,  and  depth,  with  debris  having  13,400  particles  to  the  gram,  or  grade  (B) — Con. 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
1.32 

Ft.'/sec. 
0.3U3 

Om.  /sec. 

Om./sec. 
162 
173 
257 
261 

Minutes. 

5 
3 
3 

Per  cent. 

Per  cent. 
1.64 
1.63 
1.90 
2.12 

Feet. 
16 
16 
16 
16 

Feet. 
0.092 
.082 

Feet. 

Free. 
Do. 
Do. 
Do. 

do 

[Antidunes] 

do 

.540 

33 

40 
50 

82 
82 

29 
55 
54 
89 
92 
105 
104 
147     ' 
218 
204 
278 
361 

5 
4 
4 
4 
5 
5 
4 
4 
3 
4 
3 
3 

O.C9 
.57 
.68 

.47 
.54 
.61 
.76 
.76 
.75 
.84 
.9S 
1.28 
1.49 
1.49 
1.85 

12 
14 
14 
16 
16 
14 
16 
16 
16 
16 
16 
16 

.205 
.162 
.154 
.139 
.141 
.148 
.158 
.141 
.106 

0.207 
.165 
.152 

Dunes  

Contracted. 
Do, 
Do. 
Do. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do 

do  

Smooth  .  .  . 

do 

Antidunes.  .  . 
...do... 

d^ 

[Antidunes]  
do 

.734 

26 

30 
39 
55 
56 
63 
90 
129 
136 
134 
144 
167 
178 
182 
204 
255 
268 
280 
341 
342 
383 
485 
516 

5 
7 
6 
6 
5 
5 
6 
6 
5 
4 
4 
5 
5 
4 
3 
3 
3 
3 
3 
2 
2 
3 

.33 

.31 
.41 
.50 
.50 
.53 
.58 
.87 
.67 
.78 
.85 
.90 
.96 
.92 
1.01 
1.10 
1.14 
1.17 
1.35 
1.45 
1.51 
1.65 
1.66 

12 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.289 
.226 
.219 
.195 
.206 
.199 
.174 
.170 
.172 
.165 
.173 
.169 
.166 

.267 

Contracted. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

Transition  .  .  . 

do 

...do... 

do 

Smooth  

do 

..do... 

do 

Transition.  .  . 

do 

do 

do  

do  .  .       .          

do 

do     

...do... 

do 

.  .do  ... 

do 

1.96 

.363 

8.1 

7.5 
18 
25 
36 
40 
44 
59 
66 
68 
96 
137 
147 
177 
186 
214 
249 

8 
8 
12 
23 
12 
8 
4 
8 
6 
5 
3 
3 
3 
3 
3 
3 

.47 

.31 
.65 
.62 
.59 
.77 
.92 
1.01 
1.03 
1.07 
1.21 
1.45 
1.48 
1.62 
1.64 
1.81 
2.00 

16 
16 
16 
14 
14 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.162 
.104 
.110 
.070 
.087 
.080 
.059 
.068 
.078 
.062 
.063 
.044 
.050 

Contracted. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

47 

Smooth          

Transition  

80 



Transition... 
do  

Antidunes  
do  

[Antidunes]  

do  

do..  . 

.060 

do  

.545 

; 

53 
59 
70 
72 
90 
89 
94 
106 
144 
217 
264 
299 
307 
303 
356 

23 
0 
5 
18 
3 
5 
4 
4 
4 
4 
3 
3 
3 
3 
3 

.54 
.70 
.69 
.71 
.78 
.80 
.98 
.97 
1.12 
1.39 
1.60 
1.58 
1.54 
1.61 
1.73 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.132 
.127 
.130 
.126 
.091 
.111 
.115 

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

$6 
75 

Smooth  

109 

Transition  

...do  

do  

.099 

do  

.095 

Antidunes  

[Antidunes]  

Antidunes  

[Antidunes]  
Antidunes  

.734 

7.7 
38 
35 
101 
89 

7.6 
40 
40 
66 
88 
98 
103 
186 
185 
198 
239 
332 
341 
352 
348 
469 
520 

6 
5 
8 
8 
4 
9 
6 
3 
3 
3 
4 
3 
3 
3 
3 
2 
2 

.31 

.46 

.23 
.27 
.44 
.53 
.59 
.64 
.56 
.80 
1.05 
1.03 
1.05 
1.40 
1.39 
1.38 
1.45 
1.64 
1.87 

10 
16 
12 
14 
16 
14 
14 
16 
16 
16 
16 
16 
16 
16 
16 
16 
Ifi 

.250 
.188 
.171 
.161 

.257 
.196 

Contracted. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

.44 

.139 
.148 

Antidunes  

..do  

245 

[Antidunes)  

.150 

Antidunes  

.146 
"".\U 

Antidunes  
[Antidunes]  
Antidunes  
[Antidunes]  
do  

42  TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 

TABLE  4  (B).— Observations  on  load,  slope,  and  depth,  with  debris  having  13.400  particl-es  to  the  gram,  or  grade  (B) — Con. 


Width. 

Dis- 
charge. 

! 

Load.                                         Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feel. 
1.96 

Ft.'lsec. 
1.119 

Gm.lsec. 
6.5 
U 
29 
53 
112 

Gm.lsec. 
6.4 
13 
32 
59 
109 
110 
129 
191 
230 
317 
310 
356 
355 
406 
418 

Minutes. 
10 
10 

11 

4 
8 
6 
5 
4 
4 
3 
2 
2 
2 
3 
3 

Per  cent. 
0.21 
.28 
.37 

Per  cent. 
0.19 
.18 
.32 
.28 
.59 
.61 
.65 
.72 
.73 
.91 
1.00 
.93 
.98 
1.13 
1.31 

Feet. 
16 
.  16 
16 
12 
16 
16 
12 
16 
16 
16 
16 
16 
16 
16 
16 

Feet. 
0.411 
.353 
.345 
.229 
.315 
.211 
.188 
.188 
.184 
.187 
.176 

Feet. 
0.406 
.375 
.337 

Contracted. 
Do. 
Do. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

[Dunes) 

126 

[Smooth]        

Transition  

.145 

TABLE  4  (C). — Observations  on  load,  slope,  and  depth,  with  debris  having  5,460  particles  to  the  gram,  or  grade  (C). 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
0.44 

Ft.'/sec. 
0.093 

Gm.lsec. 

Om.lsec. 
3.2 
5.3 
9.4 
12 
19 
26 
35 
34 
46 

Minutes. 
7 
5 
5 
5 
7 
4 
5 
4 
4 

Per  cent. 

Per  cent. 
0.64 
.85 
.94 
1.01 
1.26 
1.45 
1.59 
1.71 
2.24 

Feet. 
16 
16 
16 
16 
16 
16 
16 
16 
16 

Feet. 
0.161 
.141 
.131 
.122 
.117 
.111 
.111 
.100 

Feet. 

Dunes  

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

Transition 

Smooth  .  . 

do. 

.  .do  

do. 

do  

do 

.182 

9.7 
14 
38 
43 
47 
51 
97 

6 
6 
6 
5 
4 
5 
3 

.61 
.70 
1.33 
1.32 
1.38 
1.52 
2.16 

16 
16 
16 
16 
16 
16 
16 

.235 
.233 
.168 
.176 
.172 
.153 
.138 

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

Transition 

do 

do 

do  

.66 

.093 

3.5 
8.6 
10 
15 
15 
17 
16 
22 
35 
48 
61 
74 
100 
109 
142 
146 
156 

10 
13 
7 
6 
6 
6 
6 
6 
5 
4 
4 
3 
3 
5 
3 
3 
3 

.54 
.79 
.98 
.97 
1.05 
1.05 
1.11 
1.36 
1.56 
1.98 
2.38 
2.52 
3.02 
3.41 
3.73 
3.79 
4.04 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.108 
.084 
.077 
.078 
.075 
.077 
.073 
.066 
.063 
.055 
.050 
.052 

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do4. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

[Dunes]  .  . 

do  .. 

do 

do  

Transition 

do  



Antidunes  
..     .do  

Antidunes  

do  

..do  

do  

.do  

.182 

2.0 

2.3 
9.7 

420 
11 
8 
7 
7 
6 
5 
7 
5 
8 
4 
4 
4 
4 
3 
4 
3 
4 
3 
3 
3 
3 

0.29 

.24 
.54 
.64 
.76 
.72 
.94 
1.16 
1.17 
1.54 
1.57 
1.95 
2.24 
2.32 
2.33 
2.58 
2.54 
2.63 
2.82 
2.99 
3.05 
3.20 
3.45 

64 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.242 
.136 
.124 
.132 
.127 
.117 
.096 
.099 

0.238 

[Dunes].. 

Contracted. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do  .... 

16 
18 
18 
28 
39 
45 
72 
78 
105 
113 
123 
133 
146 
155 
163 
190 
205 
220 
217 
246 

.do  

do  

1.20 

.110 

[Smooth]  

1.69 

.089 

.093 

do 

Transition  .  .  . 

.062 

do  

do 

do  

do 

do 

[Antidunes]  
do 

do  

do 

do 

.363 

22 
32 
34 
37 

5 
5 
6 
4 

.44 
.64 
.70 
.61 

16 
16 
16 
16 

.241 
.222 
.211 
.213 

Dunes  

Free. 
Do. 
Do. 
Contracted. 

Transition 

do 

as 

[Transition!... 

THE   OBSERVATIONS.  43 

TABLE  4  (C). — Observations  on  load,  slope,  and  depth,  with  debris  having  5.460  particles  to  the  gram,  or  grade  (C) — Con. 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
0.66 

Ft.'/iec. 
0.363 

Qm.jsec. 
35 
37 

Gm./sec. 
35 
40 
39 
54 
78 
83 
98 
99 
97 
121 
121 
134 
161 
182 
183 
228 
241 
245 
240 
281 
292 

Minutes. 
4 
4 
5 
5 
6 
5 
3 

3 
5 
4 
4 
3 
3 
3 
3 
3 
3 
3 
3 
3 

Per  cent. 

Per  cent. 
0.63 
.66 
.67 
.87 
.99 
1.04 
1.08 
1.07 
1.08 
1.19 
1.23 
1.41 

i.n 

1.87 
1.91 
1.97 
2.04 
2.08 
2.11 
2.29 
2.57 

Feet. 
16 
16 
16 
16 
14 
16 
14 
16 
16 
14 
14 
16 
16 
16 
16 
14 
16 
16 
16 
16 
16 

Feet. 
0.216 
.212 
.205 
.186 
.174 
.167 
.168 
.173 
.164 
.174 
.168 
.161 

Feet. 

Transition  
[Transition]  
Transition 

Contracted. 
Do. 
Free. 
Do. 
Do. 
Do. 
Contracted. 
Do. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

Smooth  . 

Ho 

.do...    . 

82 
89 

92 

Transition 

do 

do 

do.. 

do.. 

.165 

do 

do    .  . 

do 

do.. 

.545 

2.8 

2.8 
36 
45 
48 
49 
47 
41 
47 
53 
76 
88 
117 
115 
121 
130 
151 
152 
162 
196 
239 
241 
252 
275 
346 

410 
7 
4 
4 
4 
4 
7 
8 
8 
4 
3 
3 
3 
3 
3 
5 
4 
5 
4 
4 
3 
3 
3 
3 

0.15 

.20 
.54 
.51 
.56 
.57 
.58 
.60 
.60 
.57 
.78 
.82 
.96 
.98 
.99 
1.01 
1.12 
1.12 
1.21 
1.53 
1.56 
1.64 
1.69 
1.73 
2.04 

56 
16 
16 
14 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
14 
14 
16 

.579 
.312 
.288 
.284 
.278 
.288 
.303 
.308 
.300 
.255 
.260 
.223 
.223 
.228 
.242 
.226 
.239 
.222 
.234 

0.577 

[Dunes]... 

Contracted. 
Free. 
Contracted. 
Do. 
Do. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Contracted. 
Do. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

44 
46 
46 
45 

Transition 

[Transition]. 

Transition  
do 

do... 

.do... 

111 
119 
117 

do 

[Transition]  . 

do  

do 

[Smooth] 



Antidunes  
[  Antidunes]  
do  ... 

.734 

67 
67 
67 
152 
164 

61 
64 
67 
168 
163 

4 
4 
4 
3 
3 

.62 
.62 
.58 
.99 
1.04 

14 
14 
16 
14 
16 

.327 
.318 
.320 
.263 
.277 

[Transition] 

Contracted. 
Do. 
Do. 
Do. 
Do. 

.  .do... 

do 

.do      ... 

Transition 

1.00 

.182 

7.6 
11 
16 
17 
18 
26 
41 
54 
56 
70 
84 
97 
106 
114 
128 
131 
145 
166 

9 

12 
9 
5 
5 
10 
8 
6 
11 
3 
6 
4 
4 
4 
4 
4 
3 
3 

.54 
.57 
.73 
.75 
.74 
.91 
1.10 
1.31 
1.32 
1.52 
1.68 
1.85 
1.96 
2.07 
2.24 
2.32 
2.48 
2.66 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.117 
.116 
.110 
.107 
.108 

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

.do    . 

do 

.do 

do 

28 
39 
53 
53 

79 

do 

do 

do 

Smooth 

.065 
.062 

.do  

do 

105 

.051 
.065 

.do  

do 

do 

[Antidunes] 

.363 

1.9 
6.5 
19 

1.9 
5.1 
18 
24 
30 
39 
46 
48 
48 
58 
61 
83 
81 
86 
84 
86 
104 
104 
124 
131 
147 

94 
41 
10 
8 
6 
7 
6 
4 
4 
6 
5 
8 
3 
9 
3 
5 
5 
5 
3 
3 
5 

.17 
.26 
.42 

.19 
.24 
.42 
.44 
.52 
.65 
.73 
.70 
.73 
.82 
.83 
.95 
.97 
.93 
.96 
.97 
1.09 
1.09 
1.25 
1.26 
1.38 

48 
48 
48 
16 
16 
40 
16 
16 
16 
16 
16 
32 
16 
32 
16 
16 
16 
16 
16 
16 
16 

.322 
.266 
.209 
.187 
.164 
.168 
.142 
.143 
.141 
.144 
.133 
.135 

.326 
.275 
.221 

Contracted. 
Do. 
Do. 
Free. 
Do. 
Contracted. 
Free. 
Contracted. 
Do. 
Free. 
Do. 
Contracted. 
Free. 
Contracted. 
Free. 
Do. 
Do. 
Do. 
Contracted. 
Do. 
Free. 

[Dunes]   

do 

do 

42 

.61 

.170 

Transition 

46 
43 

do 

do 

85 
75 
85 
75 

.123 

do 

.133 
.121 
.133 
.114 
.114 
.115 

do 

do 

.do 

140 
158 

[Smooth] 

.do 

Smooth... 

44  TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER, 

TABLE  4  (C). — Observations  on  load,  slope,  and  depth,  with  debris  having  5.460  particles  to  the  gram,  or  grade  (C) — Con. 


Width. 

Dis- 
charge. 

Load. 

Slope.                                Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
1.00 

Ft.'lsec. 
0.363 

Gm.lsec. 

Gm.lsec. 
166 
233 
257 
256 
255 
349 
362 

Minutes. 
5 
4 
3 
3 
4 
3 
3 

Per  cent. 

Per  cent. 
1.50 
1.99 
1.98 
2.10 
2.11 
2.48 
2.65 

Feet. 
16 
16 
16 
16 
16 
16 
16 

Feet. 

Feet. 

Ant  id  unes  

do 

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do 

[  Antid  ones]  
do 

do 

.do  

.545 

48 
57 
68 
64 
71 
107 
107 
150 
153 
165 
169 
173 
181 
205 
266 
315 
292 
2% 
281 
355 
391 
398 
431 

5 
5 
4 
4 
5 
5 
5 
3 
4 
4 
5 
3 
3 
4 
4 
3 
4 
4 
3 
3 
3 
3 
3 

.51 
.53 
.61 
.62 
.62 
.83 
.88 
1.02 
1.05 
1.09 
.00 
.12 
.12 
.28 
.50 
.60 
.70 
.75 
.87 
.95 
.94 
.97 
2.17 

16 
16 
16 
16 
16 
16 
16 
14 
16 
16 
16 
14 
14 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

0.211 
.206 
.193 
.189 
.193 
.176 
.175 
.167 
.173 
.163 
.160 
.166 
.160 
.148 
.136 

Free. 
Do. 
Contracted. 
Do. 
Free. 
Do. 
Do. 
Contracted. 
Free. 
Do. 
Do. 
Contracted. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

[Dunes]                   

60 

61 

do                     

do 

146 

do                  

do 

do        

do 

158 
156 

[Smooth]                

do 

Antidunes  
do 

do     ...           

.do  '.          

do 

.do  

do 

.734 
1.119 

1.9 
2.6 

2.6 

1.6 
2.1 
3.0 
13 
15 
14 
39 
84 
85 
117 
77 
95 
132 
194 
181 
230 
267 
308 
309 
329 
444 
501 
118 
129 
348 
370 

60 
170 
180 
12 
20 
12 
8 
4 
4 
5 
4 
6 
4 
3 
6 
4 
4 
4 
3 
3 
3 
3 
3 
3 
2 
2 

0.11 
.11 
.16 
.21 
.17 
.26 

.13 
.16 
.14 
.21 
.27 
.33 
.34 
.58 
.58 
.57 
.61 
.59 
.81 
.99 
1.01 
1.15 
1.29 
1.43 
1.50 
1.51 
1.78 
2.07 
.57 
.66 
1.17 
1.22 

48 
32 
40 
56 
40 
76 
36 
16 
14 
36 
14 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
14 
14 
16 
16 

.581 

0.  579 
.570 

[Dunes] 

Contracted. 
Do. 
Do. 
Free. 
Contracted. 
Free. 
Do. 
Contracted. 
Do. 
Do. 
Free. 
Do. 
Do. 
Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Free. 
Do. 
Do. 
Contracted. 
Do. 
Do. 
Do. 

.do.  ...          

.537 

do 

13 

.459 
.420 
.322 
.232 
.241 
.261 
.249 
.233 
.221 
.194 
.203 
.187 
.199 

.467 
.413 

[Thmps] 

.do  

42 
74 
76 
103 

do 

Transition  

do 

[Transition]  

Transition  
.do... 

do 

160 

.  ...do.                   

Transition 

do 

...  .do  

do 

111 
114 
325 
325 

.312 
.305 
.256 
.253 

[Transition]          .     ... 

[Smooth] 

Smooth.  . 

1.32 

.182 

11 

9.6 
7.8 
13 
21 
30 
43 
58 
99 
120 
126 
130 
141 

10 
9 
6 
5 
6 
5 
5 
5 
5 
4 
4 
3 

.66 

.66 
.70 
.84 
.98 
1.14 
1.24 
1.64 
2.08 
2.23 
2.10 
2.29 
2.34 

16 
12 
14 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.093 
.100 
.090 
.076 
.067 
.067 
.055 
.040 
.049 
.056 
.051 
.045 

.099 

Contracted. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do 

do 

....do       

do 

Smooth  ^. 

do 

.363 

8.8 
10 
26 
27 
38 
51 
60 
85 
111 
118 
130 
129 
154 
158 
190 
237 
264 
266 
300 
308 

6 
6 
6 
6 
5 
5 
5 
5 
6 
5 
4 
4 
4 
4 
3 
3 
3 
3 
3 
3 

.33 
.31 

.55 

.32 
.35 
.56 
.56 
.72 
.79 
.89 
.07 
.23 
.27 
.32 
.33 
.46 
.44 
.80 
1.92 
2.14 
2.18 
2.30 
2.46 

16 
16 
16 
16 
16 
16 
16 
16 
14 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.163 
.171 
.140 
.127 
.123 
.122 
.120 
.106 
.104 
.100 
.097 
.085 
.090 
.089 

.174 
.170 

.178 

Contracted. 
Do. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

7.8 
26 

do 

[Dunes]  

Dunes 

do 

do 

Transition 

Smooth 

do 

..  do 

...do... 

do 

Transition 

do 

do 

THE   OBSERVATIONS.  45 

TABLE  4  (C). — Observations  on  load,  slope,  and  depth,  with  debris  having  5.460  particles  to  the.  gram,  or  grade  (C) — Con. 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surfcce. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
1.32 

Ft.'jstc. 
0.545 

Gm.fsec. 

Gm./sec. 
36 
55 
79 
96 
118 
159 
210 
248 
256 
340 
362 
407 
394 

Minutes. 
6 
6 
5 
5 
5 
5 
5 
4 
3 
3 
3 
3 
3 

Per  cent. 

Per  cent. 
0.56 
.62 
.69 
.83 
.85 
1.10 
1.37 
1.57 
1.58 
1.89 
1.95 
2.08 
2.20 

Feet. 
12 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

Feet. 
0.186 
.172 
.152 
.152 
.141 
.132 
.118 

Feet. 

Dunes    . 

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do 

do 

Transition  .             

Smooth 

do.      ...             .... 

Antidunes 

...do 

do 

do 

.734 

4.3 

4.7 
14 
29 
53 
58 
74 
105 
113 
120 
159 
227 
252 
309 
368 
405 
432 
4f>6 

120 

0.18 

.16 
a  .20 

48 

.409 
.324 

0.416 

(Dunes)  

Contracted. 
Do. 
Do. 
Do. 
Free. 
Contracted. 
Free. 
Contracted. 
Do. 
Free. 
Do. 

Do. 
Do. 
Do. 
Do. 
Do. 

...    do 

0.35 

.292 

do... 

o  .51 

.210 

do 

6 

.50 
o.55 

14 

.214 
.189 

Dunes 

5 

.65 
o.75 

16 

.188 
.117 

Transition 

o  .73 

.183 

4 
4 

4 

3 
3 
3 
3 
3 

.87 
1.05 
1.15 
1.37 
1.62 
1.72 
1.82 
1.90 

16 
16 
16 
16 
16 
16 
16 
16 

.167 
.163 
.149 
.143 

Smooth. 

[Smooth] 

Smooth                            .  . 

Transition  .  .             

[Antidunes]              

do.. 

do  .... 

1.96 

.363 

11 

9.5 
16 
26 
40 
60 
77 
90 
124 
163 
175 
203 
227 
227 

10 
6 
10 
7 
4 

.45 

.35 
.59 
.71 
.90 
.99 
1.09 
1.21 
1.41 
1.63 
1.67 
1.85 
1.93 
2.03 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.128 
.108 
.105 
.100 
.089 
.080 
.078 
.067 
.063 
.064 
.067 
.061 
.060 

.141 

[Dunes]...                .  .  . 

Contracted. 
Free. 
Contracted. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

Dunes  

25 

.73 

.113 

do. 

Transition.. 

do. 

do  

Smooth 

do  

Transition 

do... 

do.... 

do. 

do  

.545 

30 
55 
79 
101 
138 
165 
210 
241 
2«6 
321 
331 
355 
386 

8 
5 
5 
6 
5 
4 
4 
4 
4 
3 
3 
3 
3 

.56 
.70 
.79 
.92 
1.07 
1.10 
1.34 
1.38 
1.58 
1.85 
1.85 
1.96 
2.06 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.148 
.127 
.113 
.115 
.100 
.098 
.094 
.094 
.080 
.071 

Dunes  

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

...do 

do..                     .      . 

Transition  .  .             

Smooth. 

do  

Transition 

do.... 

do 

Antidunes 

do.... 

do. 

[Antidunes]  

.734 

36 
36 

32 
41 
49 
55 
63 
73 
72 
87 
101 
96 
166 
241 
259 
302 
334 
350 
362 
368 

8 
10 
5 
5 
6 
7 
5 
4 
6 
5 
3 
4 
4 
3 
3 
3 

0 

3 

.42 

.54 

.45 
.41 
.37 
.49 
.60 
.58 
.61 
.60 
.70 
.75 
.92 
1.17 
1.22 
1.31 
1.53 
1.49 
1.45 
1.50 

16 
18 
16 
12 
16 
16 
16 
16 
16 
16 
14 
16 
•16 
16 
16 
14 
16 
16 

.205 
.194 
.177 
.171 
.149 
.165 
.155 
.155 
.120 
.147 
.139 
.117 
.113 
.106 
.105 

.195 
.209 

[Dunes]  

Contracted. 
Do. 
Free. 
Do. 
Contracted. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

Dunes 

do... 

do  

70 
68 

.53 
.46 

.161 
.175 

[Dunes]. 

Dunes  

do 

Transition 

121 

.67 

.145 

do 

Dunes       .  .               

Transition  
Smooth  .  . 

[Smooth] 

Transition 

Antidunes.               

[Antidunes] 

...do  

do 

1.119 

90 
112 
131 
138 
140 
149 
158 
215 
2.53 
331 
3"iO 
397 

5 
6 
4 
6 
5 
5 
4 
5 
3 
3 
3 
3 

.53 
.61 
.64 
.56 
.65 
.69 
.71 
.78 
.78 
.97 
1.03 
1.12 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.222 
.188 
.184 
.180 
.196 
.186 
.189 
.178 
.171 
.166 
.152 

.205 
.202 
.180 

Free. 
Contracted. 
Do. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

117 
130 
133 

.68 
.60 
.65 

Smooth 

do..                       .  .  . 

129 
132 

.79 
.79 

.199 
.192 

Smooth 

do 

..  do 

do 

.    do 

do  

Transition 

a  Computed  graphically  from  data  in  a  notebook  afterward  lost. 


46  TBANSPORTATION   OF   DEBBIS  BY   BUNKING   WATER. 

TABLE  4  (D). — Observations  on  load,  slope,  and  depth,  with  debris  having  1,460  particles  to  the  gram,  or  grade  (D). 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
0.66 

Ft.»/iec. 
0.093 

Om.  /sec. 

Om./sec. 
5.3 
8.1 
13 
20 
21 
29 
36 
45 
47 
62 

Minutes 
6 
6 
5 
4 
5 
4 
6 
4 
4 
4 

Per  cent. 

Per  cent. 
0.80 
.94 
1.19 
1.39 
1.68 
1.83 
1.88 
1.98 
2.25 
2.47 

Feet. 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

Feet. 

0.077 
.077 
.075 
.076 
.063 
.058 
.061 
.063 
.056 
.056 

Feel. 

Dunes.  .  . 

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do 

Transition  .  .  . 

do 

Smooth  .  . 

do  

.    do.   . 

do                            

do 

....do  

.182 

6.4 

5.8 
14 
17 
23 
30 
47 
53 
83 
108 
121 
127 
134 

7 
6 
6 
6 
5 
4 
4 
8 
9 
4 
4 
4 

.39 
.77 
.82 
.95 
1.10 
1.26 
1.40 
1.95 
2.10 
2.2fi 
2.28 
2.39 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

.166 
.147 
.135 
.123 
.105 
.108 
.108 
.090 
.090 
090 
.093 
.090 

Contracted. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do 
Do. 
Do. 
Do. 

.do                          

...do  



Transition  
..do  

...do...                    

do    .                       

...do...             

do                           

..  do.    .             

do                              

.545 

8.9 
8.9 
18 
32 
33 

6.9 
9.5 
17 
27 
37 
64 
92 
145 
172 
203 
209 
218 
231 
331 
310 

6 
5 
5 
5 
4 
5 
4 
4 
4 
4 
4 
3 
3 
3 
3 

.30 
.28 
.61 
.61 

.19 
.20 
.51 
.57 
.64 
.81 
.96 
1.25 
1.42 
.55 
.58 
.57 
.65 
.98 
2.02 

16 
16 
14 
16 
16 
12 
16 
16 
16 
12 
12 
12 
14 
12 
12 

.460 
.422 
.403 
.318 
.316 
.267 
.237 
217 
.211 
.202 
.209 
.191 
.177 
.196 

0.464 
.425 
.382 
.322 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Free. 
Do. 
Do 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 



Transition  

do  . 

do 

do.       .           

do 

do                            

do 

do                         

1.00 

.182 

9.5 
16 
22 
27 
28 
48 
49 
56 
80 
98 
109 
126 
152 

7 
5 
5 
5 
4 
4 
4 
4 
4 
4 
4 
4 
3 

.69 
.84 
1.06 
1.08 
1.14 
1.34 
1.39 
1.57 
1.83 
2.09 
2.33 
2.55 
2.75 

14 
16 
14 
16 
16 
16 
16 
16 
16 
16 
14 
14 
16 

.110 
.100 
.094 
.099 
.088 
.086 
.082 
.071 
.066 
.070 
.066 
.062 
.064 

Dunes...             
do 

Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do 
Do 
Do 
Do 
Do 
Do 
Do 

Transition 

do 

do 

do                 

do 

do 

do 

do 

do 

.363 

0 
1.8 
1.8 
5.9 
12 
12 
23 

0 
2.3 
2.0 
5.4 
12 
12 
23 
28 
35 
86 
112 
130 
141 
170 
181 
-229 
258 

.037 

40 
23 
24 
24 
20 
20 
24 
16 
16 
16 
16 
14 
16 
14 
16 
14 
14 

.418 
.292 
.283 
.245 
.224 
.235 
.209 
.170 
.164 
.136 
.120 
.116 
.117 
.115 
.103 
.104 
.101 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

81 
66 
20 
22 
26 
12 
6 
5 
4 
4 
4 
3 
3 
4 
4 
3 

.16 

.19 
32 
.27 
.56 
.53 

.18 
.18 
.25 
.35 
.43 
.58 
.65 
.80 
1.14 
1.32 
1.44 
1.49 
1.65 
1.75 
1.94 
2.11 

.289 
.286 
.252 
.224 
.229 
.199 

[Dunes]  ... 

do 

Dunes 

[DunesJ 

do 

do 

Dunes 

Transition 

do 

do 

do 

do 

do 

1.85 
2.00 

.099 
.098 

do 

do 

.545 

30 
32 

30 
31 
66 
143 
168 
229 
256 
281 
310 
340 

4 

\ 
4 

4 
3 
3 
3 
3 

.53 

.53 
.55 
.78 
.11 
.26 
.58 
.61 
.62 
.74 
.91 

16 
16 
16 
16 
16 
16 
16 
14 
14 
14 

.236 
.241 
.187 
.166 
.162 
.136 
.136 
.140 
.136 
.136 

.240 

Contracted. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do 

do 

do 

do 

do 

do  

THE   OBSERVATIONS.  47 

TABLE  4  (D). — Observations  on  load,  slope,  and  depth,  with  debris  having  1,460  particles  to  the  gram,  or  grade  (D) — Con. 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
1.00 

Fl.'/tcc. 
0.734 

Gm.lstc. 
0 
5.8 
19 
20 
49 

Qm.lsec. 
0 
6.5 
12 
21 
56 
66 
82 
10S 
170 
189 
193 
265 
293 
354 
377 

Minutei. 

Per  cent. 
0.085 

Per  cent. 

Feet. 
40 
36 
24 
36 
16 
12 
16 
14 
14 
16 
16 
16 
16 
16 
14 

Feet. 

Feet. 

Dunes  forming  

Contracted. 
Do. 
Do. 
Do. 
Do. 
Free. 
Contracted. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

29 
18 
15 
4 
4 
3 
4 
3 
4 
4 
3 
3 
3 
3 

.18 
.22 
.26 

0.21 
.32 
.37 
.61 
.61 
.75 
.83 
.98 
1.09 
1.19 
1.39 
1.46 
1.73 
1.76 

0.491 
.423 
.413 
.285 
.250 
.241 
.220 
.206 
.198 
.195 
.189 
.185 
.175 
.162 

0.482 
411 
.410 

[Dllnfts] 

do.' 

do 

do  

83 

Transition                 .      ... 

Smooth  

do 

..do... 

do                   

...do... 

do    . 

do  

1.32 

.363 

8.9 
10 
25 
59 
60 
100 
102 
154 
166 
169 
195 
202 

11 
11 
23 
53 
57 
89 
97 
135 
136 
161 
236 
247 

10 
10 
8 
5 
2 
4 
3 
3 
3 
3 
3 
3 

.50 
.40 
.46 

.30 
.34 
/  .67 
1.00 
1.05 
1.25 
,1.25 
ll.54 
1.57 
1.67 
1.89 
1.91 

16 
16 
16 
16 
16 
16 
16 
16 
14 
14 
16 
16 

.137 
.171 
.137 
.124 
'.129 
.107 
.111 
.090 
.094 
.089 
.088 
.087 

.140 
.171 

.154 

[Dunes] 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Free. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do                 

(  Dunes]  
Dunes  

.  do  

[Smooth]            

Smooth  .  .  . 

do 

...do  

.734 

17 
22 
48 
50 
53 
51 
65 
68 
86 
90 
99 
172 
202 
192 
192 

15 
25 
43 

49 
47 
60 
72 
59 
72 
83 
84 
170 
174 
209 
200 

10 
10 
10 
8 
8 
5 
5 
5 
5 
5 
4 
3 
3 
3 
3 

.28 
.33 
.60 

.51 
.64 

.34 
.37 
.52 
.58 
.61 
.62 
.55 
.70 
.79 
.67 
1.15 
1.00 
.98 
1.15 
1.17 

16 
16 
14 
16 
16 
14 
16 
16 
16 
16 
16 
12 
14 
12 
12 

.360 
.272 
.243 
.224 
.233 
.202 
.201 
.206 
.184 
.181 

.354 
.304 
.227 
.248 
.231 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do                    

[Dunesl 

do    

.75 
.66 

.228 
.192 

1.28 
1.00 
.97 

[Smooth]            .  .  ........ 

.154 
.160 
.156 
.152 

.164 
.168 

do 

do  . 

do                       

48  TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 

TABLE  4  (E). — Observations  on  load,  slope,  and  depth,  with  debris  having  142  particles  to  the  gram,  or  grade  (E). 


Width. 

Dis- 
charge. 

Load  . 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

B£sr 

Feet. 
0.66 

Ft.'lsec. 
0.182 

Om.lsec. 
25 
25 
31 
31 
31 

Gm./sec. 
21 
21 
28 
31 
30 

Minutes. 
15 
19 
15 
13 
15 

Per  cent. 

Per  cent. 
1.20 
1.27 
1.43 
1.48 
1.56 

Feet. 
16 
16 
16 
16 
16 

Feet. 
0.125 
.117 
.115 
.115 
.114 

Feet. 

Contracted. 
Do. 
Do. 
Do. 
Do. 

Transition 

.363 

38 

38 
48 
48 
48 

31 

31 
42 
46 

47 

9 

9 
9 
10 
10 

1.04 

1.04 

1.27 
1.26 
1.29 

16 

10 
16 
16 
16 

.200 

Transition     (dunes     to 
smooth). 

Contracted.' 

Do. 
Do. 
Do. 

.197 
.200 
.185 

Transition 

.734 

48 
48 
96 
90 

47 
50 
95 
97 

6 
6 
4 
4 

1.03 
1.09 
1.43 
1.46 

16 

16 
16 
16 

.372 
.387 

Contracted. 
Do. 
Do. 
Do. 

do 

do 

do 

1.119 

50 
91 
91 
101 
101 
101 
101 
193 
193 
193 

44 
80 
85 
94 
87 
94 
95 
167 
184 
186 

6 
3 
3 
3 
3 
3 
3 
3 
3 
3 

.56 
1.23 
1.31 
1.11 
1.24 
1.31 
1.29 
1.38 
1.28 
1.41 

14 

16 
16 
16 
16 
16 
16 
16 
16 
16 

.562 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

do 

do 

do 

1.00 

.182 

6.4 
19 
19 
47 
50 
50 
78 

5.2 
19 
19 
44 
44 
50 
75 

21 
10 

16 
8 
8 
9 
5 

.59 
1.11 
1.12 
1.80 
1.80 
1.94 
2.42 

16 
16 
16 
16 
16 
16 
16 

Contracted.  (?) 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

.094 
.087 
.081 
.077 
.093 

.363 

0.065 

32 
32 
'    48 
48 
48 
32 
16 
16 
16 
16 

.357 
.314 
.241 
.242 
.218 
.212 
.161 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Free. 
Contracted. 
Do. 
Do. 

Tr. 

.11 

Dunes  formin? 

2.1 
2.1 
6.7 
6.6 
24 
50 
95 
142 

2.9 
2.2 

7.7 
7.9 
22 
53 
104 
168 

61 
CO 
20 
20 
34 
9 
5 
4 

.23 

.22 
.25 
.44 

.44 
.87 
1.28 
1.83 
2.29 

0.277 

""."215" 

.208 
.107 

[Dunes]  

Dunes  

.46 
.39 
.80 
1.24 
1.92 
2.26 

Transition  .  .  . 

do 

.113 
.110 

do 

.115 

do  

.734 

.04 

32 
32 
48 
48 
32 
40 
28 
16 
16 
36 
36 
16 
16 
16 
16 
16 
16 

.618 
.562 
.447 
.442 
.411 
.403 
.364 
.303 
.301 
-     .324 
.326 
.273 
.236 
.223 
.187 
.189 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

Tr. 

.07 

2.1 
2.1 
6.9 
6.9 
21 
21 
24 
41 
41 
47 
87 
95 
142 
142 
203 

1.7 
3.  C 
4.5 
10 
20 
20 
26 
44 
47 
45 
86 
100 
135 
150 
220 

70 
63 
31 
25 
8 
18 
16 
10 
6 
8 
6 
4 
3 
3 
2 

.19 

.18 
.29 
.26 
.46 
.48 
.47 
.73 
.74 
.83 
1.19 
1.23 
1.46 
1.50 
1.61 

do 

.17 
.31 
.31 
.43 

.459 
.403 
.407 
.363 

do 

...do     . 

do 

...do.... 

do 

.61 

Dunes  

do 

.82 

Dunes 

Transition 

1.17 
1.50 
1.47 
1.80 

Smooth  

1.119 

47 
47 
189 
189 

50 
50 
170 
190 

6 
8 
3 
3 

.52 
.59 
1.15 
1.18 

16 
16 
16 
16 

.380 
.406 

Contracted. 
Do. 
Do. 
Do. 

1.32 

.363 

20 
20 
86 

20 
20 
93 

15 
10 

5 

.67 
.73 
1.78 

16 
16 
16 

.131 
.129 
.097 

Contracted. 
Do. 
Do. 

....do  

.734 

33 
86 
86 

33 
85 
86 

9 
4 
3 

.60 
1.10 
1.12 

16 
16 
16 

.  I'll!) 
.198 
.211 

Dunes  

Contracted. 
Do. 
Do. 

1.119 

50 
50 
172 
172 

46 
50 
163 
171 

6 
4 
3 
3 

.58 
.62 
1.13 
1.22 

16 
16 
16 
16 

.324 
.317 

Contracted. 
Do. 

do 

THE    OBSERVATIONS.  49 

TABLE  4  (F). — Observations  on  load,  slope,  and  depth,  with  debris  having  22.1  particles  to  the  gram,  or  grade  (F). 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

ctio'r  **w. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
0.66 

Ft.'/sec. 
a.  182 

Qm.litt. 

11 
58 
58 

Gm.jsec. 

12 
52 
53 

Minutes. 
17 
23 
11 
8 

Per  cent. 

Per  cent. 
1.29 
1.31 
2.51 
2.50 

Feet. 
16 
16 
16 
16 

Feet. 
0.118 
.129 
.102 
.106 

Feet. 

Contracted. 
Do. 
Do. 
Do. 

[Smooth] 

Smooth  .  .                  

do  

.363 

26 
26 
71 
71 

26 
26 
80 
71 

17 
12 
6 
10 

1.12 
1.13 
1.89 
1.96 

16 
16 
16 
16 

.207 
.204 
.176 
.170 

Smooth 

Contracted. 
Do. 
Do. 
Do. 

do 

.7:(1 

35 
35 

106 
106 

37 
38 
103 
108 

9 
6 
3 
3 

.97 
1.00 
1.68 
1.75 

16 

16 
16 
16 

.346 
.340 

Contracted. 
Do. 
Do. 
Do. 

ghines]  
unes 

do  

1.00 

.182 

6.8 
41 
41 

6.8 
42 
44 

21 

7 
7 

1.36 
2.49 
2.53 

16 
16 
16 

.090 
.078 
.080 

Free.  (?) 
Do. 
Do. 

do                      

do 

.363 

10 
10 
51 
51 
51 
120 

9.5 
10 
47 
51 
56 
120 

18 
20 
14 
10 
8 
3 

.85 
.91 
1.65 
1.70 
1.68 
2.47 

16 

16 
16 
16 
16 
16 

.160 
.161 
.128 
.130 
.140 

Smooth 

Free.  (?) 
Do. 
Do. 
Do. 
Do. 
Do. 

do            

[Smooth]  

do    

.734 

26 
26 
104 
104 

25 
25 
98 
114 

14 

12 
5 
5 

.77 
.77 
1.52 
1.60 

16 
16 
16 
16 

.268 
.265 
.217 
.211 

Dunes    .  . 

Contracted. 
Do. 
Do. 
Do. 

do                    

...do   .. 

do                       

1.119 

52 
52 
207 
207 

53 
56 
197 
209 

5 
5 
3 
3 

.80 
.85 
1.65 
1.67 

16 
16 
16 
16 

.330 
.343 

Free.  (?) 
Do. 
Do. 
Do. 

..do.... 

...do  

do  

1.32 

.363 

21 
21 
70 
70 

21 
21 
68 
72 

7 
21 
9 
5 

1.16 
1.21 
2.05 
2.07 

16 
16 
16 
16 

.116 
.118 
.114 
.108 

Smooth  

Contracted. 
Do. 
Do. 
Do. 

do  

do  

.734 

26 
26 
26 
104 
104 

26 
27 
27 
102 
106 

13 

12 
12 
3 
4 

.83 
.85 
.86 
1.48 
1.58 

16 
16 
16 

16 
16 

.212 
.209 
.215 
.176 
.180 

Dunes  

Contracted. 
Do. 
Do. 
Do. 
Do. 

do  

[Dunes]  

1.119 

58 
58 
58 
212 
212 

50 
50 
59 
188 
208 

12 

9 
6 
2 
2 

.74 
.78 
.84 
1.55 
1.60 

16 
16 
16 
ID 
16 

.275 
.288 
.284 

Dunes  

Contracted. 
Do. 
Do. 
Do. 
Do. 

do  

do                

g>unes]  

unes  

TABLE  4  (G). — Observations  on  load,  slope,  and  depth,  with  debris  having  5.9  particles  to  the  gram,  or  grade  (G). 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
0.66 

Ft.'/sec. 
0.363 

Gm.liec. 
10 
25 
25 
51 
50 
100 
100 

Om.  /sec. 
11 
25 
28 
47 
50 
100 
105 

Minutes. 
15 
12 
15 
6 
8 
3 
3 

Per  cent. 

Per  cent. 
1.11 
1.44 
1.48 
1.82 
1.90 
2.56 
2.70 

Feet. 
16 
16 
16 
16 
16 
16 
16 

Feet. 
0.198 
.186 
.192 
.175 
.175 
.160 
.158 

Feet. 

Contracted  . 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

.734 

10 
10 
25 
50 
50 
105 
203 
210 

8.5 
11 
26 
48 
49 
104 
210 
219 

14 
23 
11 
7 
11 
5 
3 
3 

.68 
.70 
.95 
1.19 
1.19 
1.71 
2.43 
2.35 

16 
16 
16 
16 
16 
16 
16 
14 

.373 
.364 
.342 
.322 
.324 
.292 
.261 
.264 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

1.119 

10 
50 
50 
100 
100 
203 
203 
304 
304 

12 
49 
54 
94 
114 
212 
215 
311 
331 

18 
6 
6 
4 
4 
3 
3 
3 
3 

.62 
.98 
1.02 
1.35 
1.32 
1.97 
1.95 
2.40 
2.36 

16 
16 
16 
16 
16 
16 
16 
16 
16 

.558 
.460 
.451 
.414 
.411 
.374 
.378 
.354 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

l  

! 

| 

'  

20021°— No.  80—14- 


50 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


TABLE  4  (G). — Observations  on  load,  slope,  and  depth,  with  debris  having  5.9  particles  to  the  gram,  or  grade  (G) — Con. 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

•a.,                DlS- 

!ed-        tance. 

By  gage. 

Br{,er 

Feet. 
1.00 

Ftysec. 
0.363 

Gm.lsec. 
10 
20 
25 
25 
34 
51 
102 

Om./sec. 
10 
21 
25 
28 
34 
50 
97 

Minutes. 
20 
15 
12 
18 
12 
9 
5 

Per  cent. 

Per  cent. 
1.27 
1.48 
1.61 
1.62 
1.76 
2.09 
2.74 

Feet. 
16 
16 
16 
16 
16 
16 
16 

Feet. 
0  143 
.139 
.136 
.141 
.132 
.129 
.114 

Feet. 

Smooth   .              

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

Smooth 

do 

..  do... 

do 

.734 

10 
20 
25 
25 
50 
101 
201 
201 

11 

20 
25 
26 
49 
100 
189 
212 

42 
15 
15 
11 
10 
5 
4 
3 

.78 
.86 
.95 
.97 
1.27 
1.69 
2.30 
2.37 

16 
16 
16 
16 
16 
16 
16 
16 

.272 
.248 
.248 
.251 
.235 
.214 
.190 
.191 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

[Smooth] 

do   



(Smooth]  

1.119 

10 
25 
25 
50 
50 
101 
102 
204 
306 
306 

10 
24 
25 
53 

54 
100 
103 
214 
297 
310 

20 
15 
17 
6 
7 
4 
4 
3 
3 
3 

.64 
.66 
.67 
.97 
.90 
1.22 
1  31 

16 
16 
16 
16 
16 
16 
16 
16 
14 
14 

.389 
.357 
.359 
.324 
.326 
.308 
.307 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

1.78 
2.04 
2.21 

.275 
.252 
.253 

1.32 

.363 

25 
25 
51 

51 

98 
98 

24 
26 
49 
47 
95 
123 

12 
9 
10 
10 
3 
4 

1.90 
1.97 
2.25 
2.34 
3.02 
3.10 

16 
16 
16 
16 
16 
16 

.115 
.104 
.108 
.105 
.093 
.097 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 

.734 

10              9 
10            12 
25            24 
25            27 
51            52 
102          102 
203  '        200 

21 
18 
13 
13 
6 
4 
3 

.82 
.82 
1.08 
1.14 
1.41 
1.82 
2.44 

16 
16 
16 
16 
16 
16 
16 

.210 
.210 
.194 
.200 
.193 
.171 
.163 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

1.119 

10 
51 
51 
98 
98 
197 
295 
304 

14 
46 
67 
95 
110 
200 
282 
302 

19 
4 
6 
3 
4 
3 
3 
3 

.71 
.90 
1.07 
1.31 
1.40 
1.83 
2.17 
2.26 

16 
16 
16 
16 
16 
16 
16 
16 

.297 
.270 
.261 
.248 
.245 
.230 
.201 
.202 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

TABLE  4  (H). — Observations  on  load,  slope,  and  depth,  with  debris  having  2.0  particles  to  the  gram,  or  grade  (JJ). 


Width. 

Dis- 
charge. 

Load  . 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

Bv  pro- 
files. 

Feet. 
0.66 

Ft.*lsec. 
0.363 

Gm.lsec. 
10 
10 
21 
21 
52 
52 

Om.lsec. 
9.2 
12 
17 
22 
51 
56 

Minutes. 
21 
16 
18 
14 
5 
6 

Per  cent. 

Per  cent. 
1.49 
1.58 
1.80 

Feet. 
16 
16 
16 
16 
16 
16 

Feet. 
0.184 
.184 
.183 
.173 
.167 
.173 

Feet. 

Contracted. 
Do. 
Do. 
Do. 

1.84 
2.43 
2.47 

.734 

10 
10 
21 
21 
52 
52 
105 
209 
209 

7.9 
11 
20 
22 
51 
52 
105 
209 
222 

18 
17 
17 
15 
8 
8 
4 
3 
3 

.90 
.95 
1.10 
1.19 
1.50 
1.51 
2.02 
2.69 
2.92 

16 
16 
16 
16 
16 
16 
16 
16 
16 

.345 
.348 
.333 
.344 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

.320 
.310 
.289 
.250 
.253 

1 

1.119 

10 
10 
10 
26 
52 
52 
52 
105 
105 
105 
209 
209 

12 
10 
12 
26 
53 
54 
53 
98 
106 
113 
209 
209 

16 
20 
21 
15 
13 
10 
9 
3 
6 
3 
3 
3 

.74 
.81 
.89 
1.03 
1.26 
1.28 
1.33 
1.65 
1.63 
1.62 
2.31 
2.38 

16 
16 
16 
16 
16 
16 
16 
14 
16 
16 
16 
16 

.502 
.510 
.503 
.470 
.442 
.437 
.447 
.391 
.398 
.392 

| 

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

.340 
.334 

THE    OBSERVATIONS.  51 

TABLE  4  (I). — Special  group  of  observations  on  load,  slope,  and  depth,  with  debris  of  grade  (f);  for  discussion  of  form  ratio. 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

(  haracter  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water         ,,    , 
surface. 

Dis- 
tance, 

»™^-  B&r- 

Feet. 
1.00 

Ft.'/sec. 
0.734 

(im./sec. 
74 
76 
160 

Om./sec. 
84 
45 
194 

Minutes. 
4 
4 
3 

Per  cent. 

Per  cent. 
0.58 
.58 
.99 

Feel. 
16 
14 
16 

Feet. 
0.232 
.241 
.194 

Feet. 

Transition 

Contracted. 
Do. 
Do. 

do  

.923 

96 
94 
254 

103 
103 
280 

3 
3 

2 

.58 
.62 
1.09 

16 
16 
16 

.270 
.282 
.233 

Contracted. 
Do. 
Do. 

do 

1.119 

111 
114 

118 
129 

3 
3 

.57 
.66 

14 
14 

.312 
.305 

Transition 

Contracted. 
Do. 

1.20 

.734 

66 
67 
193 
199 

74 
74 
218 
223 

3 

3 
3 
3 

.51 
.60 
1.05 
1.07 

16 
16 
16 
16 

.222 
.215 
.173 
.175 

Transition  

Contracted. 
Do. 
Do. 
Do. 

Transition  

{Transition] 

.923 

99 
249 
246 

104 
266 
274 

3 
3 
3 

.65 
1.08 
1.01 

14 
16 
16 

.232 
.210 
.201 

Contracted. 
Do. 
Do. 

...do... 

An 

1.021 

105 
110 
304 

107 
109 
325 

3 

4 
3 

.57 
.54 
1.20 

16 
16 
16 

.250 
.262 
.197 

[Transition] 

Contracted. 
Do. 
Do. 

Transition  .  . 

1.119 

108 
111 

322 
339 

114 
123 
292 
326 

4 
4 
3 
3 

0.56 

.62 
.61 
1.00 
1.03 

12 
16 
16 
14 

.269 
.269 
.224 
.231 

.277 

[Transition]  

Contracted. 
Do. 
Do. 
Do. 

Transition  , 

[Smooth]  

Smooth 

1.40 

.734 

76 
76 
220 
216 

74 
75 
215 
226 

4 
4 
3 
3 

.63 
.59 
1.04 
1.07 

16 
14 
16 
16 

.199 
.196 
.158 
.151 

Contracted. 
Do. 
Do. 
Do. 

.      .do  

.do  

.923 

72 
82 
271 
283 

79 
74 
279 
297 

4 
3 
3 

.51 
.57 
1.03 
1.04 

16 
16 
16 
14 

.235 
.237 
.176 
.175 

[Transition]  . 

Contracted. 
Do. 
Do. 
Do. 

Transition 



Smooth   . 

[Smooth] 

1.021 

82 
292 
289 

97 
319 
323 

\ 
3 

.58 
1.03 
1.08 

16 
14 

16 

.248 
.181 
.180 

Contracted. 
Do. 
Do. 

[Smooth]  
Smooth  

1.119 

105 
123 
292 
284 
292 

104 
130 
325 
337 
312 

4 
4 
3 
3 
3 

.57 
.61 
.97 
1.01 

1.07 

16 
16 
14 
14 
16 

.254 
.231 
.213 
.215 
.203 

Contracted. 
Do. 
Do. 
Do. 
Do. 

(Smooth] 

Smooth  .  . 

1.60 

.734 

69 
265 
269 

70 
236 
268 

4 
3 
3 

.60 
1.14 
1.12 

12 
16 
16 

.191 
.121 
.122 

Contracted. 
Do. 
Do. 

Smooth  .  .  . 

do 

.923 

67 
69 
67 
298 
289 

82 
69 
66 
299 
280 

4 
5 
4 
3 
3 

.45 
.46 

.50 
1.05 
1.07 

12 
12 
12 
16 
16 

.233 
.202 
.224 
.161 
.161 

Contracted. 
Do. 
Do. 
Do. 
Do. 

do  

[Dunes] 

Smooth  .  .  . 

do 

1.021 

67 
63 
66 
310 

66 
73 
80 
311 

4 
4 
4 
3 

.39 
.47 
.51 
1.06 

12 

12 
12 
16 

.243 
.244 
.232 
.169 

Contracted. 
Do. 
Do. 
Do. 

1.119 

82 
96 
85 
336 

83 
91 
98 
332 

4 
4 
4 
3 

.49 
.47 
.55 
.99 

16 
12 
12 
12 

.227 
.234 
.233 
.188 

Transition 

Contracted. 
Do. 
Do. 
Do. 

[Transition]  

do 

1.80 

.734 

69 
70 
211 
205 

69 
72 
204 
203 

4 
4 
3 
8 

.61 
.60 
1.04 
1.06 

16 
16 
16 
16 

.174 
.174 
.137 
.130 

Contracted. 
Do. 
Do. 
Do. 

do 

....do  

.923 

70 
67 
260 
260 

75 
74 
258 
273 

4 
4 
3 
3 

.53 
.58 
1.04 
1.04 

14 

14 
16 
16 

.206 
.204 
.157 
.152 

Dunes  

Contracted. 
Do. 
Do. 
Do. 

do         ... 

[Smooth]  

1.021 

70 
69 
70 

292 
296 

69 
76 
76 
323 

308 

4 
4 
4 
3 
3 

.51 

.52 
.57 
1.00 
1.01 

16 
14 

16 
14 
16 

.224 
.227 
.225 
.155 
.158 

Contracted. 
Do. 
Do. 
Do. 
Do. 

g)unes] 

Smooth  .  .  . 

do 

1.119 

64 
75 
386 
363 

66 
74 
375 
385 

4 
.  4 
3 
3 

.42 
.47 
1.05 
1.05 

14 

14 
16 
12 

.240 
.243 
.160 
.167 

Dunes  
do 

Contracted. 
Do. 
Do. 
Do. 

An 

52  TRANSPORTATION    OF   DEBRIS   BY   RUNNING   WATER. 

TABLE  4  (I). — Special  group  of  observations  on  load,  slope,  and  depth,  with  dfbrin  of  grade  (0) — Continued. 


Width. 

Dis- 
charge. 

Load. 

Slope. 

Depth. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Water 
surface. 

Bed. 

Dis- 
tance. 

By  gage. 

By  pro- 
files. 

Feet. 
1.96 

FtJ/sec. 
0.734 

Om.jsec. 
85 
84 
234 
222 
234 

Gm.jsec. 
83 
88 
236 
231 
250 

Mimttes. 
4 
4 
3 
3 
3 

Per  cent. 

Per  cent. 
0.61 
.69 
1.05 
1.09 
1.12 

Feet. 
16 
16 
16 
16 
16 

Feet. 
0.147 
.146 
.117 
.123 
.122 

Feet. 

[Transition] 

Contracted. 
Do. 
Do. 
Do. 
Do. 

do 

.923 

96 
246 

97 
279 

3 
3 

.62 
1.02 

16 
16 

.176 
.143 

Transition  

Contracted. 
Do. 

1.021 

84 
85 
94 
328 
316 

85 
87 
104 
343 
352 

4 
4 
3 
3 
3 

.50 
.57 
.54 
1.09 
1.09 

16 
14 
16 
16 
16 

.194 
.200 
.193 
.141 
.144 

[Transition] 

Contracted. 
Do. 
Do. 
Do. 
Do. 

[Transition]  

do                           '. 

Transition  

1.119 

82 
85 
82 
345 
328 

82 
84 
94 
341 
363 

4 
4 
4 
3 
3 

.45 
.57 
.52 
1.00 
1.00 

12 
16 
16 
16 
16 

.214 
.222 
.219 
.150 
.147 

Contracted. 
Do. 
Do. 
Do. 
Do. 

Transition 

do 

Smooth  

(Smooth! 

TABLE  4  (J). — Observations  on  load,  slope,  and  depth,  with  debris  of  two  or  more  grades  mixed. 


Designation  of  mixture,  component  grades, 
and  percentages  by  weight. 

Width  of 
trough. 

Dis- 
charge. 

Load. 

Slope  of  bed. 

Depth  by 
gage. 

Character  of 
bed. 

Feed. 

Collec- 
tion. 

Period. 

Per  cent. 

Dis- 
tance. 

CAif.WCAISO  *  (C}50 

Feet. 
1.00 

Ft.»lsec. 
0.363 

Om.lsec. 
42 
90 
128 
171 

Qm.liec. 
43 
91 
140 
169 

Minutes. 
8 
4 
4 
3 

0.58 
.92 
1.23 
1.47 

Feet. 
16 
16 
16 
16 

Feel. 

Transition. 
Do. 
Antidunes. 
Do. 

(A3Gi)=(A)75  '  (G)25                                    

1.00 

.363 

20 
90 
80 
175 

175 

IN 
74 
89 
164 
176 

12 
5 
4 
3 
3 

.51 
.93 
.95 
1.38 
1.36 

16 

16 
16 
16 
16 

Smooth. 
Transition. 
Do. 
Antidunes. 
Do. 

(AjGi)«=(A)67  •  (G)33 

1.00 

.363 

20 
42 
42 
87 
110 

20 
42 
42 
88 
119 

12 
6 
6 
3 
3 

.53 
.62 
.64 
1.00 
1.18 

16 

16 
16 
16 
16 

Smooth. 
Do. 
Do. 
Antidunes. 
Do. 

(AiGi)—  (A)50  '  (G)50  

1.00 

.363 

22 
46 
91 

22 
43 
93 

8 
6 
4 

.68 
1.01 
1.42 

16 
16 
16 

Smooth. 
Transition. 
Do. 

(AiG2)—  (A)33  •  (G)67 

1.00 

.363 

42 

40 

6 

1.36 

16 

Smooth. 

(  \iG<)—  (A)22  •  (G)78 

1.00 

.363 

22 
43 

21 
42 

20 

7 

1.30 
1.79 

16 
16 

(B«Fi)=(B)78  •  (F''22  

1.00 

.363 

42 
84 
113 
113 
169 

43 
92 
105 
125 
159 

6 
3 
3 
3 
2 

.59 
.86 
.99 
1.05 
1.38 

16 
16 
16 
16 
16 

Smooth. 
Do. 
Do. 
Do. 

Transition. 

(BjFi)=-(B)64  •  (F)36 

1.00 

.363 

39 
78 
105 
157 

42 
80 
109 
157 

7 
5 
4 
2 

.57 
.85 
1.11 
1.49 

16 
16 
16 
16 

Smooth. 
Do. 
Do. 
Do. 

(BiFi)=(B)47  *  (F)53                                

1.00 

.363 

45 
90 
120 
181 

49 
95 
130 
162 

7 
4 
3 
2 

.73 
1.14 
1.41 
1.61 

16 
16 
16 
16 

Transition. 
Do. 
Smooth. 
Do. 

(BiFz)—  (B)31  •  (F)69                       

1.00 

.363 

28 
41 

81 

35 
40 

82 

11 
6 
4 

.86 
1.06 
1.56 

16 
16 
16 

Dunes. 
Do. 
Do. 

(BjF()—  (B)18  •  (F)82                              

1.00 

.363 

16 
27 
40 

17 
29 

43 

15 
10 

6 

.82 
1.16 
1.46 

16 
14 
16 

Dunes. 
Do. 
Do. 

(CiEi)—  (C)79  •  (E)21                             .     ... 

1.00 

.363 

43 
85 
115 
171 

49 
84 
118 
171 

8 
4 
3 
2 

.70 

16 
16 
16 
16 

Smooth. 
Do. 
Do. 
Do. 

.93 
1.15 
1.52 

THE   OBSERVATIONS.  53 

TABLE  4  (J). — Observations  on  load,  slope,  and  depth,  with  debris  of  two  or  more  grades  mixed — Continued. 


Designation  of  mixture,  component  grades, 
and  percentages  by  weight. 

Width  of 
trough. 

Dis- 
charge. 

Load. 

Slope  of  bed. 

Depth  by 
gage. 

Character  of 
bed. 

Feed. 

Collec- 
tion. 

Period. 

Per  cent. 

Dis- 
tance. 

(C!E1)-(C)65  :  (E)35  

Feet. 
1.00 

Ftflsec. 
0.182 

Gm.jtec. 
78 
104 

Gm./sec. 
81 
105 

Minutes. 
3 
4 

1.77 
2.06 

Feet. 
16 
16 

Feet. 

Smooth. 
Do. 

.363 

43 
85 
114 
155 
171 
155 
171 

42 
85 
117 
170 
169 
170 
173 

7 
4 
3 
3 
3 
4 
3 

.69 
1.03 
1.27 
1.57 
1.65 
1.60 
1.74 

16 
16 
16 
16 
16 
16 
16 

Transition. 
Do. 
Smooth. 
Do. 
Do. 
Do. 
Do. 

(CiEj)—  (C)48  '  (E)52 

1.00 

.182 

26 
40 
40 
53 
51 
53 
51 
81 
77 
81 

28 
40 
41 
50 
60 
52 
61 
67 
78 
87 

12 
10 
10 

6 

.99 
1.17 
1.27 
1.43 
1.50 
1.51 
1.55 
1.82 
1.95 
1.89 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

0.096 
.090 
.091 
.081 

Transition. 
Do. 
Do. 
Smooth. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

6 

.082 

4 

5 

.363 

39 
39 
78 
104 
104 
104 
157 

40 
41 
78 
93 
115 
119 
158 

6 
7 
5 
3 
3 
3 
3 

.74 
.76 
1.07 
1.37 
1.40 
1.43 
1.85 

16 
16 
16 
16 
16 
16 
16 

.150 
.151 

Transition. 
Do. 
Smooth. 
Do. 
Do. 
Do. 
Do. 

(CiEj)=(C)31  •  (E)69 

1.00 

.182 

26 
39 
52 
78 

26 
39 
57 

76 

10 
8 
6 
3 

1.10 
1.37 
1.64 
2.22 

16 
16 
16 
16 

.093 
.088 
.082 

Transition. 
Do. 
Smooth. 
Do. 

.363 

39 
39 

78 
108 
157 

39 
39 
81 
112 
158 

8 
9 
3 
3 
3 

.77 
.84 
1.25 
1.54 
2.10 

16 
16 
16 
16 
16 

Dunes. 
Do. 
Do. 
Transition. 
Do. 

(CiE()-(C)19  :  (E)81. 

1.00 

.182 

20 
26 
39 
39 
52 
52 

20 
28 
40 
42 
51 
54 

14 
10 
6 
6 
6 
12 

1.02 
1.29 
1.61 
1.65 
1.88 
1.93 

16 
16 
16 
16 
16 
16 

Dunes. 
Do. 
Do. 
Do. 
Transition. 
Do. 

.085 
.079 
.084 
.078 
.082 

.363 

40 
53 
78 
109 

40 
56 
76 
109 

8 
7 
4 
3 

1.00 
1.14 
1.45 
1.75 

16 
16 
16 
16 

Dunes. 
Do. 
Transition. 
Do. 

(C(Gi)—  (C)80  :  (O)20 

1.00 

.363 

31 
93 
140 
187 
187 

32 
92 
140 
194 
195 

11 
5 
4 
3 
3 

.62 
1.07 
1.30 
1.48 
1.62 

16 
16 
16 
16 
16 

~Dunes. 
Transition. 
Do. 
Smooth. 
Do. 

(CjGi)-(C)67  :  (G)33  

1.00 

.363 

31 
31 
31 
61 
92 
92 
122 
183 

29 
29 
31 
66 
90 
95 
120 
180 

.59 
.63 
.62 
.90 
1.03 
1.08 
1.20 
1.65 

16 
16 
16 
16 
16 
16 
16 
16 

Smooth. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 
Do. 

12 
10 
7 
4 
5 
3 
2 

(CiG,)=(C)50  :  (G)50  

1.00 

.363 

17 
50 
100 
133 
133 

18 
50 
105 
128 
145 

15 
10 
3 
3 
3 

.71 
.99 
1.34 
1.62 
1.66 

16 
16 
16 
16 
16 

Smooth. 
Do. 
Do. 
Do. 
Do. 

(CiGj)-(C)33  :  (G)67...  . 

1.00 

.363 

16 
32 
32 

64 

15 
34 
36 

69 

16 
10 
10 

7 

.95 
1.17 
1.27 
1.59 

16 
16 
16 
16 

Transition. 
Smooth. 
Do. 
Transition. 

(E4Gi)—  (E)SO  :  (Q)20  

1.00 

.363 

15 
31 
61 

17 
31 

58 

15 
10 

8 

.72 
'  1.00 
1.39 

16 
16 
16 

Transition. 
Smooth. 
Transition. 

(E,G])-(E)C7  :  (G)33  

1.00 

.363 

16 
32 
63 

15 
30 

59 

18 
11 
6 

.76 
1.03 
1.45 

16 
16 
16 

Smooth. 
Do. 
Transition. 

(EiGi)-(E)50  :  (G)50  

1.00 

.363 

16 
31 

62 

18 
34 
60 

18  1           .87 
10            1.12 

s         i.eo 

16 
16 
16 

Smooth. 
Do. 
Transition. 

54  TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 

TABLE  4  (J). — Observations  on  load,  slope,  and  depth,  with  debris  of  two  or  more  grades  mixed — Continued. 


Designation  of  mixture,  component  grades, 
8nd  percentages  by  weight. 

Width  of 
trough. 

Dis- 
charge. 

Load.                             Slope  of  bed. 

Depth  by 
gage. 

Character  of 
bed. 

Feed. 

Collec- 
t.on- 

Period. 

Per  cent. 

Dis- 
tance. 

(EiGi)—  (E)33  :  (O)67 

Feet. 
1.00 

FtJjsec. 
0.363 

Gm./sec. 
16 
23 
30 
30 
15 
45 

Gm./sec. 
15 
23 
29 
32 
49 
54 

Minutes. 
14 
12 
12 
11 
7 
7 

0.95 
1.18 
1.35 
1.33 
1.73 
1.74 

Feet. 
16 
16 
16 
16 
16 
16 

Feet. 

Transition. 
Smooth. 

Transition. 
Do. 

(A!CiG!)=(A)25  :  (C)25  :  (G)50 

1.00 

.363 

16 
16 
16 
32 

126 
126 

13 

17 
14 
33 
88 
131 
151 

15 
10 
15 
10 
5 
4 
4 

.77 
.79 
.82 
.88 
1.32 
1.59 
1.67 

16 
16 

16 
16 
16 
16 
16 

Transition. 
Do. 
Smooth. 
Do. 
Do. 
Transition. 
Do. 

(CDEFG)-(C)45  :  (D)35  :  (E)12  :  (F)6  :  (G)2. 

1.00 

.182 

21 
42 
84 

25 
45 
84 

12 
10 
5 

.82 
1.23 
1.72 

14 
16 
16 

Dunes. 
Transition. 

.363 

67 
67 
84 
84 
84 
113 
113 
169 
169 

69 
69 
80 
83 
83 
120 
118 
172 
173 

8 
6 
5 
5 
5 
5 
5 
4 
4 

.81 
.85 
.91 
.93 
.95 
1.18 
1.19 
1.53 
1.54 

16 
16 
16 
16 
16 
16 
16 
16 
16 

Smooth. 
Do. 
Do. 
Do. 
Do. 

.545 

84 
84 
169 
169 
253 
253 

90 
92 
151 
177 
228 
242 

6 
5 
3 
3 
3 
3 

.65 
.67 
1.07 
1.13 
1.40 
1.37 

16 
16 
16 
16 
16 
16 

Smooth. 
Do. 
Do. 
Do. 
Do. 

TABLE  4  (K). — Observations  on  load,  slope,  and  depth,  with  unassorted  debris.  a 


Width. 

Dis- 
charge. 

Load. 

Slope  of  bed. 

Depth 
by  gage. 

Character  of  bed. 

Outfall. 

Feed. 

Collec- 
tion. 

Period. 

Per  cent. 

Distance. 

Feet. 
1.00 

Ftfjsec. 
0.182 

Gm./sec. 
19 
19 
38 
75 
145 

Gm.isec. 
'    18 
18 
38 
76 
147 

Minnies. 
15 
15 
8 
3 
2 

0.74 
.79 
1.00 
1.55 
2.28 

Feet.    • 
16 
16 
16 
16 
16 

Feet. 

Dunes  

Contracted. 
Do. 
Do. 
Do. 
Do. 

-do 

Smooth  .  .  . 

do  

do  

.363 

22 
22 
38 
38 
77 
154 

20 
27 
34 
42 
74 
150 

10 
12 
6 
6 
5 
3 

.43 

.50 
.59 
.67 
.89 
1.31 

16 

16 
16 
16 
16 
16 

0.169 
.176 
.157 
.158 

Transition  

Contracted. 
Do. 
Do. 
Do. 
Do. 
Do. 

do  

Smooth  

...do.... 

do  

...do.... 

a  The  very  coarsest  particles  were  removed  by  passing  the  sample  through  a  6-mesh  sieve,  and  the  very  finest  by  passing  it  over  a  60-mesh  sieve 
It  retained  the  equivalents  of  grades  (A),  (B),  (C),  (D),  (E),  and  (F). 


CHAPTER  II.— ADJUSTMENT  OF  OBSERVATIONS. 


OBSERVATIONS   ON   CAPACITY  AND  SLOPE. 

THE    OBSERVATIONAL    SERIES. 

Each  of  the  experiments  in  stream  traction 
involved  six  quantities — (1)  a  fineness,  or 
grade  of  d6bris,  (2)  a  width  of  trough,  (3)  a 
discharge,  (4)  a  slope,  (5)  a  load,  or  capacity, 
and  (G)  a  depth  of  current.  The  experiments 
were  arranged  in  series,  for  each  of  which  grade, 
width,  and  discharge  were  constant,  while 
within  each  the  magnitudes  of  slope,  capacity, 
and  depth  were  varied.  There  will  be  frequent 
occasion  to  mention  these  secondary  units  of 


the  experimental  work,  and  whenever  the  title 
series  seems  not  sufficiently  specific  they  will  be 
called  observational  series.  The  number  of 
such  series  recorded  in  Table  4  is  153. 

The  factors  of  grade,  width,  and  discharge, 
which  are  related  to  an  individual  series  as 
fixed  conditions,  or  constants,  do  in  fact  assume 
the  character  of  variables  when  series  is  com- 
pared with  series;  but  their  modes  of  determi- 
nation and  combination  are  not  of  such  char- 
acter that  their  numerical  values  can  be 
checked  and  adjusted  by  means  of  recorded 
relations. 


300 


Jzoo 


(0 

a. 
1) 
o 


100 


0  I  2 

Slope 

FIGURE  13. — Plot  of  a  single  series  of  observations  on  capacity  and  slope.    Capacity  ingramsofdebrispersecond.     Slope  in  percent.    Themodesof 

traction  are  indicated. 


In  each  experiment  the  values  of  slope,  load, 
and  depth  are  mutually  dependent ;  within  each 
series  they  form  a  triple  progression,  the  depth 
decreasing  while  slope  and  load  increase; 
but  the  laws  of  these  interdependent  varia- 
tions are  partly  masked  by  irregularities  in  the 
sequences.  As  a  preliminary  to  the  general 
discussion,  the  observational  values  were  sub- 
jected to  a  process  of  adjustment,  whereby  the 
sequences  were  freed  from  irregularities.  The 
irregularities  are  made  manifest  by  the  com- 
parison of  the  sequences  of  two  variables,  and 
first  consideration  will  be  given  to  those  of 
capacity  and  slope. 

Figure  13  exhibits  the  relations  of  capacity 
to  slope  as  observed  in  a  single  series  of  experi- 


ments (that  for  grade  (C),  with  w=l.32  feet 
and  Q  =  0.363  ft.3/sec.).  The  ordinates  indicate 
capacity,  as  measured  by  debris  delivered 
at  the  lower  end  of  the  trough;  the  abscissas 
represent  slope,  as  measured  on  the  bed  of 
the  channel.  The  arrangement  of  the  observa- 
tional dots  suggests  that  if  the  observations 
were  harmonious  the  dots  would  fall  in  a 
line  of  simple  curvature.  Such  a  line  would 
express  the  law  connecting  capacity  and  slope. 
The  departures  of  the  dots  from  such  linear 
arrangement  represent  irregularities,  or  errors, 
in  the  experimental  data.  The  adjustment 
proposed  is  the  replacement  of  the  imperfectly 
alined  dots  by  a  generalized  or  representative 
line,  or  the  replacement  of  the  inharmonious 

55 


56 


TRANSPORTATION    OF    DEBRIS    BY    RUNNING    WATER. 


values  of  capacity  and  slope  by  a  system  of 
harmonious  or  adjusted  values. 

ERRORS. 

As  a  first  step  in  the  treatment  of  the  errors 
of  the  data  they  were  studied  with  a  view  to 
the  discrimination  of  the  systematic  and  the 
accidental. 

The  three  modes  of  traction — the  dune,  the 
smooth,  the  antidune — although  intergrading, 
are  mechanically  different.  It  was  surmised 
that  they  might  differ  in  efficiency,  so  that  the 
capacity-slope  curve  might  show  a  step  in 


400 


200 


100 
5-80 

|  60 

40 


20 


7 


.4  .6        .8       I  23 

Slope 

FIGURE  14. — Logarithmic  plot  of  a  series  of  observations  on  capacity 
and  slope.    Compare  figure  13. 

passing  from  one  to  another;  and  it  was  also 
surmised  that  the  law  connecting  capacity 
with  slope  might  not  be  the  same  for  the 
several  modes.  A  suggestive  observation  had 
shown  that  on  very  low  slopes — slopes  so  low 
that  capacity  is  minute — the  current  changes 
an  artificially  smoothed  bed  of  debris  to  a 
system  of  dunes,  and  that  with  the  develop- 
ment of  dunes  the  load  is  notably  increased 
without  any  change  in  general  slope.  To  test 
the  surmises  all  the  series  were  plotted  on  log- 
arithmic section  paper.  Figure  14  shows  a 
logarithmic  plot  of  the  same  data  which  appear 
in  figure  13;  and  it  will  be  observed  that  the 
line  suggested  by  the  points  has  much  less 


curvature  in  the  logarithmic  plot.  Its  ap- 
proximation to  a  straight  line  makes  the  study 
of  its  local  peculiarities  comparatively  easy. 
The  examination  of  the  plots,  while  not  dis- 
proving the  surmises,  showed  that  whatever 
diverse  influences  may  be  exerted  by  the  modes 
of  traction,  they  are  too  small  to  be  discrim- 
inated from  the  irregularities  due  to  other 
causes. 

Other  sources  of  systematic  error  are  con- 
nected with  the  methods  of  experimentation. 

INTAKE    INFLUENCES. 

As  the  water  entered  the  experiment  trough 
from  the  stilling  tank  it  was  accelerated,  the 
gain  in  mean  velocity  being  associated  with  a 
quick  descent  in  the  surface  profile.  Beyond 
this  descent  the  profile  usually  rose  somewhat, 
and  there  was  commonly  a  moderate  develop- 
ment of  fixed  waves.  This  development  was 
modified  and  the  waves  were  on  the  whole 
reduced  by  the  addition  of  the  debris.  As  it 
fell  into  the  water  the  debris  had  no  forward 
momentum,  and  it  therefore  tended  to  retard 
the  current.  But  the  d6bris  also  accumu- 
lated on  the  bottom,  reducing  the  depth  of  the 
water  at  that  point,  and  this  reduction  neces- 
sitated an  increase  in  mean  velocity.  In  the 
immediate  neighborhood  of  the  place  where 
debris  was  fed  the  slope  of  the  water  was 
affected  by  an  abnormality  distinguishable 
from  the  intake  abnormality  proper,  and  the 
joint  abnormality  faded  gradually  downstream. 
The  nature  of  these  features  varied  with  the 
discharge  and  load,  with  the  gradual  develop- 
ment of  the  adjusted  slope,  and  also  with  the 
mode  of  feeding.  During  the  greater  part  of 
the  experimental  work  the  feeding  was  either 
automatic  and  continuous  or  else  manipulated 
by  hand  in  such  way  as  to  make  it  nearly  con- 
tinuous, but  for  a  minor  part  the  feeding  was 
intermittent,  a  measureful  of  debris  being 
dumped  into  the  water  at  regular  intervals. 

OUTFALL    INFLUENCES. 

In  all  the  earlier  work  the  trough  had  the 
same  cross  section  at  the  lower  end  as  else- 
where, and  the  water  fell  freely  from  its  open 
end  to  the  settling  tank.  As  the  resistance  to 
its  forward  motion  was  less  at  the  outfall  than 
within  the  trough,  the  water  flowed  faster 
there.  Its  faster  flow  diminished  the  resist- 
ance just  above,  and  thus  the  influence  of 


ADJUSTMENT   OF   OBSERVATIONS. 


57 


outfall  conditions  extended  indefinitely  up- 
stream. An  expression  of  this  influence  was 
found  in  the  water  profile,  which  was  usually 
convex  in  the  lower  part  of  the  trough,  the 
degree  of  convexity  diminishing  upstream. 
Its  effect  on  the  profile  of  the  bed  is  not  readily 
analyzed,  because  that  profile  is  adjusted 
through  the  velocity  of  water  at  the  bottom  of 
the  current,  and  the  bed  velocity  is  not  simply 
related  either  to  mean  velocity  or  to  depth. 


~~A 

FIGURE  15. — Diagrammatic  longitudinal  section  of  outfall  end  of  experi- 
ment trough,  illustrating  influence  of  sand  arrester  on  water  slope. 

A  second  factor  at  the  outfall  end  was  the 
arrangement  for  separating  the  debris  from  the 
current.  This  included  a  well,  ABCD,  figure 
15,  which  was  sunk  below  the  trough  bed  and 
into  which  the  debris  sank,  while  the  current 
passed  on  to  the  outfall  at  E.  In  part  of  the 
work  the  space  AD  was  entirely  open;  in 
another  part  a  coarse  screen  was  stretched 
across  it.  In  either  case  the  resistance  of  this 
part  of  the  channel  bed  differed  from  the 

D 


resistance  along  the  debris  slope  and  may  have 
been  greater  or  less.  From  the  well  to  the 
outfall,  DE,  the  frictional  resistance  was  loss 
than  elsewhere.  As  the  fixed  part  of  the 
channel  bed,  DE,  was  horizontal  and  the 
debris  portion,  GA,  was  inclined,  the  profile 
of  the  bed  changed  at  A.  Projected  forward, 
the  slope  GA  passed  below  E  to  H,  and  when 
the  debris  slope,  down  which  the  transporting 
current  flowed,  was  steep,  the  part  DE  was 
related  to  it  somewhat  as  a  dam.  The  tend- 
ency of  the  quasi-darn  was  to  retard  the  cur- 
rent near  the  outfall  and  make  the  water 
profile  concave,  and  in  some  of  the  experiments 
the  profile  actually  became  concave.  Other 
outfall  factors  were  recognized,  but  they  are 
not  here  mentioned  because  they  are  believed 
to  be  of  relatively  small  importance. 

In  the  reduction  of  observations  on  slope  an 
attempt  was  made  to  lessen  the  effect  of  intake 
and  outfall  influences  by  omitting  from  the 
calculations  the  profile  data  obtained  near  the 
ends  of  the  trough.  The  data  from  a  con- 
siderable number  of  experiments  were  finally 
discarded  altogether  and  do  not  appear  in  the 
tables.  To  replace  the  discarded  data  experi- 
ments were  afterward  made  with  a  modified 
apparatus. 


FIGUKE  16. — Diagrammatic  longitudinal  section  of  d<!bris  bed  and  stream,  in  a  long  trough. 


CHANGES  IN  APPARATUS. 

As  the  terminal  influences  of  all  kinds 
diminish  with  distance  from  the  trough  ends,  a 
manifest  mode  of  avoidance  is  to  employ  a 
very  long  experimental  trough  and  determine 
slopes  from  observations  in  the  medial  portion 
exclusively.  A  trough  length  of  150  feet  was 
tried  and  proved  moderately  successful  for 
very  low  slopes.  For  steeper  slopes  there  were 
practical  difficulties — the  need  either  of  a 
trough  with  adjustable  slope  or  of  a  very  deep 
trough  and  a  correspondingly  large  stock  of 
assorted  debris — and  these  led  to  the  consid- 
eration of  other  plans.  The  one  adopted  was 
to  contract  the  channel  at  the  outfall  and  thus 
increase  the  resistance  to  flow  at  that  point, 
and  with  the  contracted  outfall  to  use  a  trough 
of  moderate  length.  The  theory  of  this  plan 
may  be  illustrated  by  a  diagram.  In  figure 
16,  AB  represents  in  profile  the  bed  of  a  long 


trough,  CB  the  profile  of  debris,  and  DB'  the 
water  profile.  In  the  tract  EF  the  water  and 
debris  profiles  are  nearly  parallel  and  depth  and 
velocities  are  therefore  practically  uniform. 
From  F  to  B'  the  water  profile  is  notably  con- 
vex because  the  resistance  to  flow  afforded  by 
the  water  itself  steadily  diminishes  toward  B'. 


FIGURE    17.— Diagrammatic   longitudinal    section   of  outfall  end   of 
trough,  illustrating  influence  of  contractor. 

The  plan  undertook  to  introduce  at  F,  by  con- 
traction, a  resistance  equivalent  to  that 
afforded  by  the  water  beyond  F  and  then  dis- 
pense with  the  portion  of  the  trough  between 
F  and  B.  In  the  longitudinal  section,  figure 
17,  ADB  is  the  bed  of  the  trough,  with  the  well 
for  catching  debris,  as  already  shown  in  figure 


58 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


15;  EGB  is  the  profile  of  one  side  wall.  The 
walls  converge  from  F  to  G  (see  fig.  3),  pro- 
ducing the  contraction  at  the  outfall.  It  was 
found  that  the  bed  of  debris,  instead  of  run- 
ning to  a  feather  edge  at  D  (compare  fig.  16), 
held  its  thickness  to  C'  and  ended  in  a  steep 
incline. 

The  device  of  contraction  accomplished  its 
purpose  of  avoiding  terminal  difficulties,  but  it 
was  found  to  aggravate  certain  other  difficul- 
ties, next  to  be  described. 

RHYTHM. 

Whenever  the  profile  of  a  current  was  deter- 
mined by  a  series  of  measurements  applied  to 
the  water  surface,  that  surface  was  found  to  be 
in  a  state  of  unrest.  Its  position  in  any  ver- 
tical fluctuated  upward  and  downward  rhyth- 
mically. The  amplitude  of  oscillation,  which 
might  be  great  or  small,  was  not  constant: 


that  is,  the  rhythm  was  not  simple,  but  com- 
pound. It  consisted  apparently  of  many 
rhythmic  elements  differing  one  from  another 
in  period  and  amplitude. 

The  rhythmic  quality,  thus  easily  appre- 
ciated by  watching  the  play  of  the  surface  in 
relation  to  a  fixed  point,  permeated  every 
function  of  the  current — the  slope  of  its  profile, 
both  local  and  general,  the  slope  of  its  bed,  the 
quantity  of  debris  transported,  the  mode  of  its 
transportation.  The  rhythm  of  the  dune  has 
already  been  described,  but  associated  with  the 
dunes  were  greater  debris  waves,  also  traveling 
downstream  and  each  involving  the  volume  of 
many  dunes.  In  the  bed  of  the  long  trough  a 
series  of  them  could  be  seen;  in  the  shorter 
trough  one  or  two  might  be  made  out,  or  the 
effect  might  be  only  an  alternate  temporary 
steepening  and  flattening  of  the  general  slope. 
The  rhythm  of  the  antidune  was  accompanied 


FIGURE  18. — Profiles  ol  channel  bed,  illustrating  fractional  rhythms  associated  with  dunes  of  greater  magnitude.  Scales:  Each  horizontal 
space  =20  feet;  each  vertical  space  =0.4  foot.  For  Nos.  1  to  5  the  average  slope  is  0.2  per  cent,  and  the  average  load  (of  grade  (C))  per  foot 
of  channel  width  is  7  gm./sec.  For  Nos.  6  to  9  the  average  slope  is  0.6  per  cent  and  the  average  load  37  gm./sec. 


by  rhythmic  paroxysms  and  doubtless  by 
other  rhythms  which  escaped  recognition 
because  the  steep  slopes  with  which  the  anti- 
dune  was  associated  were  not  studied  in  the 
long  trough.  Figure  18  shows  a  few  channel- 
bed  profiles  in  which  rhythmic  features  appeal 
to  the  eye,  but  the  greater  number  of  such  pro- 
files merely  show  an  irregularity  in  which 
periodicity  is  not  conspicuous.  It  is  probable 
that  the  currents  were  affected  by  numerous 
coexistent  rhythms,  which  served  to  confuse 
one  another  and  thus  masked  periodicity  except 
when  some  one  rhythm  was  stronger  than  the 
rest. 

The  condition  of  relatively  smooth  channel 
bed  which  intervened  between  the  conditions 
characterized  severally  by  dunes  and  anti- 
dunes  was  also  a  condition  of  relative  uni- 
formity in  all  the  activities  of  the  current,  and 
when  it  prevailed  the  rhythmic  variations  were 
at  a  minimum. 


The  rhythms  of  the  transporting  stream 
manifestly  constitute  a  group  of  phenomena 
worthy  of  systematic  study,  but  the  Berkeley 
laboratory,  having  a  definite  and  different 
theme,  treated  them  only  as  difficulties  inter- 
fering with  its  work.  It  sought  the  capacity 
for  load  inhering  in  the  average  of  all  the 
diverse  slopes  presented  by  the  rhythms,  and 
it  necessarily  treated  the  deviations  of  slope 
measurements  from  that  average  as  accidental 
errors. 

The  rhythms  affected  the  determinations  of 
loads  as  well  as  slopes.  The  variations  of 
profile  were  effected  by  erosion  and  deposi- 
tion, and  a  current  which  was  eroding  or 
depositing  carried  more  load  at  one  point 
than  at  another.  As  the  loads  were  largely 
determined  by  weighing  the  debris  delivered 
at  the  trough  end  in  a  limited  time,  the  amount 
obtained  would  depend  in  part  on  the  phase  of 
slope  variation  near  the  point  of  delivery. 


ADJUSTMENT    OF    OBSERVATIONS. 


59 


In  the  earlier  experiments  with  contraction 
at  outfall — as  in  other  experiments — it  was 
necessary  to  continue  a  run,  with  uniform 
discharge  and  uniform  feed  of  debris,  until 
the  slope  of  the  sand  bed  had  been  automat- 
ically adjusted  to  the  conditions.  The  criteria 
adopted  for  recognition  of  a  state  of  adjust- 
ment were  two — that  the  water  slope  equal 
the  bed  slope  and  that  the  rate  of  delivery 
of  debris  equal  the  rate  of  feed.  It  was  found 
impracticable  to  satisfy  these  tests,  because 
both  slopes  and  the  rate  of  delivery  fluctuated 
through  a  wide  range,  and  an  approximate 
adjustment,  if  attained,  could  not  be  made  to 
continue.  The  state  of  affairs  may  be  likened 
to  the  waving  of  a  flag  in  the  wind;  at  the 
outer  margin  the  amplitude  of  the  undulation 
is  much  greater  than  close  to  the  staff.  In 
the  long  trough  the  outfall  end  corresponded 
to  the  staff,  giving  a  fixed  position  and  ele- 
ment of  uniformity  to  which  the  profiles  con- 
formed, and  the  rhythmic  departures  were 
greater  with  distance  from  the  outfall.  The 
shorter  trough  when  combined  with  'end  con- 
traction represented  a  segment  of  the  long 
trough  at  a  distance  from  the  outfall  and  was 
correspondingly  subject  to  great  fluctuations. 
Despite  these  difficulties,  the  nature  of  which 
was  not  well  understood  at  the  tune,  a  large 
number  of  experiments  were  made  in  this  way. 

The  work  with  free  outfall  was  affected 
chiefly  by  terminal  influences,  and  as  these 
produced  systematic  errors  there  was  danger 
of  false  conclusions.  The  work  with  con- 
tracted outfall  was  affected  by  accidental 
errors  of  such  magnitude  as  largely  to  mask 
the  nature  of  the  laws  sought.  Between 
these  perils  of  Scylla  and  Charybdis  a  middle 
course  was  finally  steered  by  using  a  moder- 
ate amount  of  contraction,  whereby  the  recog- 
nized systematic  errors  were  practically  avoided 
without  the  introduction  of  insuperable  rhyth- 
mic irregularities. 

SLOPES    OF    DEBRIS    AND    WATER    SURFACE. 

The  slope  of  the  bed  of  debris  to  which 
measurement  was  applied  had  been  estab- 
lished by  the  stream  as  that  appropriate  to 
the  stream's  load  of  debris.  It  was  caused 
by  the  load,  in  conjunction  with  the  discharge 
and  other  conditions,  and  it  accurately  sufficed 
to  give  the  stream  capacity  for  that  load. 
This  was  my  point  of  view  in  arranging  the 


experimental  methods,  and  accordingly  one  of 
the  principal  measurements  undertaken  was 
that  of  the  debris  profile.  But  the  slope  more 
generally  considered  in  hydraulic  studies  is  that 
of  the  water  surface.  Head,  the  hydraulician's 
ordinary  measure  for  the  determination  of 
power,  is  the  vertical  interval  between  two 
points  of  the  water  surface,  and  slope  is  the 
loss  of  head  in  a  unit  of  distance.  Under 
conditions  of  uniform  flow  the  two  profiles 
are  parallel,  but  for  various  reasons  our  ex- 
perimental currents  ordinarily  lacked  so  much 
of  uniformity  that  the  two  slopes  were  appre- 
ciably different.  I  do  not  find  it  easy  to 
decide  which  slope  should  be  regarded  as  the 
true  correlative  of  capacity  for  traction,  but 
as  all  our  laboratory  data  include  the  debris 
slope,  while  the  determinations  of  water  slope 
were  relatively  infrequent,  the  discussion  of 
results  has  adhered  almost  exclusively  to  the 
former.  If  the  water  slope  is  the  true  cor- 
relative, then  the  use  of  the  debris  slope 
involves  a  systematic  error. 

THE    LOGARITHMIC    PLOTS. 

When  the  data  of  an  observational  series 
are  plotted  on  ordinary  section  paper,  as  in 
figure  13,  and  a  representative  line  is  drawn 
through  or  among  them,  that  line  is  the  graphic 
equivalent  of 

C=f(S) (2) 

When  they  are  plotted  on  logarithmic  section 
paper,  as  in  figure  14,  and  a  representative 
line  is  drawn,  that  line  is  the  graphic  equiva- 
lent of 


.(3) 


The  second  equation,  or  line,  is  the  logarith- 
mic form  of  the  first. 

As  already  mentioned,  a  logarithmic  plot 
was  made  of  each  observational  series.  The 
plot  included  primarily  the  slopes  of  the  de- 
bris bed  and  the  determinations  of  load  from 
the  delivery  of  debris  at  the  end  of  the  trough, 
but  it  included  also,  with  distinctive  nota- 
tion, such  determinations  as  were  available 
of  water-surface  slopes  and  of  load  based  on 
the  rate  of  feed  at  the  head  of  the  trough. 
The  notation  also  classified  observations  with 
reference  to  the  three  modes  of  traction  and 


60 


TRANSPORTATION    OF    DEBRIS   BY    RUNNING    WATER. 


distinguished  experiments  made  with  use  of 
the  contractor  from  those  with  free  outfall. 

A  critical  study  of  these  plots  led  to  several 
conclusions.  (1)  Judged  by  the  internal  evi- 
dence of  regularity  and  irregularity,  the  ob- 
servations of  de'bris  slope  and  water  slope 
have  about  the  same  quality.  (2)  Similarly, 
the  observations  of  load  fed  and  load  deliv- 
ered have  about  the  same  quality.  (3)  As 
already  stated,  the  law  of  sequence  is  not 
discovered  to  change  in  passing  from  one 
process  of  traction  to  another;  the  assump- 
tion of  a  continuous  law  is  the  best  practic- 
able. (4)  Except  for  very  low  slopes,  the 
results  obtained  with  the  use  of  the  con- 
tractor do  not  differ  widely  from  those  with- 
out it.  (5)  There  are  differences  between 
bodies  of  experimental  data  obtained  at  dif- 
ferent stages  of  the  work — differences  of  un- 
certain source  but  presumably  connected  with 
modes  of  manipulation — which  make  it  desir- 
able to  treat  such  bodies  separately  whenever 
practicable.  (6)  The  best  representative  line 
is  not  straight  but  curved,  and  its  curvature  is 
always  in  one  direction. 

If  the  representative  line  were  straight,  func- 
tion (3)  would  have  the  form 


log  C=log  A+nlog S. 


-(4) 


in  which  A  is  a  constant  capacity  and  n  a  ratio ; 
and  function  (2)  would  have  the  form 


C^AS". 


-(5) 


On  some  of  the  plots  the  observational  points 
are  too  irregular  to  afford  trustworthy  evidence 
of  curvature.  On  most  of  them  the  indicated 
curvature  is  slight.  From  inspection  of  the 
data  during  the  progress  of  the  experiments  it 
was  thought  that  the  true  representative  line 
would  prove  to  be  straight,  in  which  case  the 
accurate  determination  of  two  points  on  the 
line  would  suffice;  and  some  of  the  experimental 
work  was  adjusted  to  that  theory.  The  series 
giving  data  for  but  two  points  on  the  logarith- 
mic plot  of  course  furnished  no  evidence  as  to 
curvature  of  the  representative  line. 

For  all  those  cases  in  which  the  position  of 
the  best  representative  line  could  be  inferred, 
with  close  approximation,  from  the  arrange- 
ment of  the  observational  points,  a  satisfactory 
adjustment  could  be  made  by  simply  drawing 


the  line  and  then  converting  its  series  of  posi- 
tions into  figures;  but  this  procedure  would 
afford  no  control  for  the  curvature  of  repre- 
sentative lines  in  cases  where  the  observa- 
tional points  were  few  or  inharmonious.  In 
order  to  make  the  stronger  series  support  the 
weaker,  the  plan  was  adopted  of  (1)  connecting 
the  lines  with  a  formula  of  interpolation,  (2) 
correlating  the  constants  of  the  formula  with 
conditions  of  experimentation,  and  thus  (3) 
giving  deductive  control  to  the  lines  of  the 
weaker  series. 

SELECTION   OF  AN   INTERPOLATION    FORMULA. 

The  best  interpolation  formula  is  one  which 
embodies  the  true  theory  of  the  relation  to 
which  the  observations  pertain.  In  the  present 
case  the  true  theory  is  not  known,  but  there 
are  certain  conditions  which  a  theory  must 
satisfy,  and  these  may  be  used  as  criteria  in 
the  selection  of  a  form  for  empiric  formulas 
of  interpolation.  Subject  to  these  criteria, 
the  form  selected  should  serve  to  minimize  the 
discrepancies  between  observed  and  adjusted 
values. 

For  the  study  of  the  character  of  the  curve 
to  represent  best  the  logarithmic  plot  of  ob- 
servations, the  data  for  de'bris  of  grade  (G) 
were  selected.  The  experiments  furnishing 
those  data  were  all  performed  by  one  method, 
the  method  using  moderate  contraction  of  the 
trough  at  outfall;  and  for  that  grade  the  ap- 
parent curvature  of  the  logarithmic  graph  is 
greater  than  for  most  others.  The  data  were 
first  plotted  (on  logarithmic  paper)  in  groups, 
each  group  containing  the  data  for  three  graphs 
which  pertain  to  the  same  width  of  channel 
but  to  different  discharges.  It  was  assumed 
that  the  three  graphs,  if  correctly  drawn,  would 
constitute  a  system,  the  one  for  the  medium 
discharge  being  intermediate  in  form  and  posi- 
tion between  the  other  two;  and  in  drawing 
them  on  this  assumption  the  forms  were  mu- 
tually adjusted.  Then  a  rearrangement  was 
made  which  grouped  together  data  agreeing  as 
to  discharge  but  differing  as  to  width  of  chan- 
nel, and  further  adjustment  was  made.  Selec- 
tion was  finally  made  of  the  graph  for  w  =  0.66 
foot  and  Q  =  0.734  ft.3/sec.,  and  this  was  drawn, 
through  the  selected  positions,  by  the  aid  of  a 
flexible  ruler.  Thus  the  curve  in  figure  19  is 
a  graphic  generalization  not  only  from  its 


ADJUSTMENT   OF   OBSERVATIONS. 


61 


particular  series  of  observations  but  from 
several  related  series.  It  was  assumed  to  be  a 
typical  or  representative  curve  for  the  func- 
tion log  ^=/i(log  8);  and  the  corresponding 


/ 

/ 

/ 

I 

/ 

1 

fr« 

1 

.-    00 

i 

8 

/ 

/ 

( 

/ 

I 

/ 

1 

.e 

i 

s 

i 

0                          'd 
Slope 

.0 

FIOUBE  19.— Logarithmic  graph  of  C=/(S),  for  grade  (Q),  M—  0.66  foot, 
Q=0.734  ft.»/sec. 

plot  on  ordinary  section  paper  (the  line  BD  in 
fig.  20)  was  assumed  to  represent  typically 
the  function  C=f(S).  The  coordinates  of  the 
line  BD  are  given  in  Table  5. 

TABLE  5. —  Values  of  capacity  for  traction,  graphically  gen- 
eralized from  data  of  Table  4  (G),  for  w=0.66  foot  and 
Q—0.734ft.3lsec.;  corresponding  to  the  curve  of  log  C= 
/!  (log  S)  in  figure  19,  and  the  curve  BD  in  figure  20. 


Slope. 

Capacity. 

Slope. 

Capacity, 

Per  cent. 

Om.fsec. 

l>er  cent. 

Q-m./sec. 

0.8 

16.3 

1.8 

124 

1.0 

31.2 

2.0 

153 

1.2 

49.9 

2.2 

1X5 

1.4 

72.0 

2.4 

220 

1.6 

97.0 

A  number  of  tentative  formulas  were  now 
compared  with  this  empiric  line,  their  param- 
eters being  computed  so  that  they  would  fit,  as 
nearly  as  practicable,  the  values  of  C  in  Table 
5.  Certain  functions,  including  the  simpler 
functions  of  the  circular  arc  and  the  exponen- 
tial function  <7=e"<s+a>,  could  not  be  fitted, 
even  approximately,  to  the  data;  but  the  fol- 


lowing   functions    yielded    curves    closely    re- 
sembling that  of  figure  20: 


(6) 

(7) 

(8) 

(9) 

.(10) 

(11) 


Functions  (6),  (7),  and  (8)  are  special  cases 
of  the  general  formula  of  interpolation  with 
integral  exponents  : 


No.  (11)  is  a  somewhat  involved  power 
function  suggested  by  results  of  a  preliminary 
discussion  of  the  laboratory  data.  Nos.  (9) 
and  (10)  are  special  cases  of  the  general  para- 
bolic function 


(z  +  a)"  = 


.(13) 


and  have  the  virtue  of  facilitating  the  graphic 
treatment  of  the  material.  Their  logarithmic 
equivalents  are,  respectively, 

log  (<7+/l)=  log  b  +  n  log  S  _______  (14) 

log  C"=log  b  +  n  log  (S-a)  _______  (15) 

and,  as  each  of  these  is  the  equation  of  a 
straight  line,  the  graphic  derivation  of  the 
exponent,  by  means  of  logarithmic  section 
paper,  becomes  a  simple  matter  after  the 
value  of  ^  or  a  has  been  determined. 

The  adjustment  of  equations  (6)  to  (11)  to 
the  specific  data  in  Table  5  gives  them  the 
following  forms,  (6a)  being  derived  from 
(6),  etc.: 

C=  -29.55  +  67.59^-7.  194S3  ______  (6a) 

C  --  12.865  +  44.08S'2  ______________  (7a) 

<7=  -  19.25  +  16.945  +  34.48S2  _______  (8a) 

(7=-10.0+41.2S"-M  _______________  (9a) 

0-70.5(5  -0.39)1*-.  .  (lOa) 


(7-31.2* 


2.68 
„«•» 


-(Ha) 


When  the  curves  corresponding  to  these 
equations  are  plotted  for  the  region  covered 
by  the  empiric  line  BD,  they  coincide  very 
closely  with  that  line.  The  greatest  departure 


62 


TRANSPORTATION    OF   DEBRIS   BY   RUNNING   WATER. 


is  in  the  curve  for  (7a),  but  its  divergence  is  not 
sufficient  to  throw  it  out  of  apparent  harmony 
with  the  series  of  points  representing  the 
original  observations. 

In  order  to  exhibit  further  the  properties  of 
the  formulas,  their  curves  were  extrapolated  in 


both  directions  from  the  locus  BD.  Table  6 
contains  the  numerical  data  used  in  plotting 
the  extensions.  Figure  20  gives  the  exten- 
sions of  the  curves  for  slopes  greater  than 
those  of  the  experiments,  and  figure  21  the 
extensions  for  smaller  slopes. 


TABLE  6. — Numerical  data  computed  for  the  construction  of  curves  in  figures  20  and  "21. 


(6a) 

(7a) 

(8a) 

(9a) 

(  lOa") 

(  lla^t 

Values  of  C  corresponding  to— 
S-0 

Gm.lsee. 
0 

Om./wc. 
0 

Om./sec. 
19  25 

Gm.lsec. 
10  0 

Om./sec. 

Qm.jsfc. 
0 

S=0  1 

845 

9  57 

S-0.2  

—.810 

—  8.30 

do 

073 

S=0.3... 

.108 

—11.07 

—  6  20 

do 

490 

S-0.4 

1.908 

—  6  97 

3  28 

039 

1  59 

S-0.5.  .. 

4.59 

—  1.96 

45 

1  93 

3  64 

S-0.6 

8.15 

3  33 

4  98 

5  54 

S—  07 

12  6 

9  50 

10  33 

S-O.B  .                

16.3 

17.9 

16  36 

16  48 

16  49 

S-3.0 

327 

358 

332 

353 

337 

323 

S—  4  0 

504 

654 

589 

631 

S-5.0 

645 

1,039 

920 

990 

849 

S—  60 

705 

1  510 

1  315 

1  422 

S-7.0.  . 

645 

2  070 

1  741 

1  531 

S—  80 

409 

2  718 

2  313 

2  519 

S-9.0  

—31 

2,916 

2  355 

1  276 

S=10.0  

3,926 

2,818 

1.400 

Per  cent. 

\          ° 

.45 

Per  cent. 
0 
29 

Per  cent. 
-1.03 

54 

Per  cent. 
-0.49 

Per  cent. 
0.39 

Per  cent. 
0 

I           8.94 
6  04 

.22 

Values  of  S  corresponding  to  point  of  inflection  

3.33 

4  37 

zpoo 


1,500 


1,000 


0  I  2  3  4  5  6  7  8 

Slope 
FIGURE  20.— Extrapolated  curves  of  f=/(S)  for  tentative  equations  of  interpolation,  and  for  slopes  greater  than  2.4  per  cent. 


The  approximate  range  of  this  series  of 
experiments  is  from  a  slope  of  0.8  per  cent  to 
one  of  2.4  per  cent.  The  extrapolated  curves 
pertain  to  slopes  from  2.4  to  10  per  cent  and 
from  0.8  to  0  per  cent.  The  prompt  divergence 
of  the  lines  as  they  leave  the  locus  to  which 
they  were  adjusted  shows  that  they  have 
widely  different  values  for  purposes  of  extra- 
polation, and  therefore  presumably  for  pur- 
poses of  interpolation. 


Attention  being  given  first  to  the  curves  for 
higher  slopes  (fig.  20),  it  will  be  observed  that 
four  of  them  ascend  with  progressively  increas- 
ing rate.  The  curve  of  formula  (1 1  a)  ascends 
continuously,  but  its  rate  of  ascent  changes 
at  the  slope  of  4.37  per  cent  from  an  increasing 
rate  to  a  decreasing  rate.  The  curve  of  formula 
(6a)  exchanges  its  increasing  rate  of  ascent  for 
a  decreasing  rate  at  the  slope  of  3.33  per  cent, 
attains  a  maximum  at  a  slope  of  about  6  per 


ADJUSTMENT   OF   OBSERVATIONS. 


63 


cent,    and   crosses    the  line   of  zero   capacity 
before  reaching  the  slope  of  9  per  cent. 

The  general  characteristics  of  stream  traction 
do  not  admit  of  a  maximum  in  the  relation 
of  capacity  to  slope.  Capacity  for  traction  is 
clearly  an  increasing  function  of  the  stream's 
velocity,  and  the  velocity  is  clearly  an  increas- 
ing function  of  the  slope.  There  is  reason  also 
to  believe  that  capacity  increases  at  an  in- 
creasing rate  up  to  the  slope  corresponding  to 
infinite  capacity.  There  are  three  forces  con- 
cerned in  traction — first,  the  force  of  the  cur- 
rent, of  which  the  direction  is  parallel  to  the 
slope;  second,  a  component  of  gravity,  when 
gravity  is  resolved  in  directions  parallel  and 


normal  to  the  slope;  third,  the  resistance  of 
the  bed,  which  is  a  function  not  only  of  the 
others,  but  inversely  of  the  slope.  Within  the 
range  of  experimental  slopes  the  component  of 
gravity  is  negligible  in  comparison  with  the 
force  of  the  current,  and  the  influence  of  slope 
on  the  resistance  is  relatively  unimportant; 
but  as  the  angle  of  stability  for  loose  material 
is  approached  the  resistance  diminishes  rapidly, 
and  at  the  slope  of  instability  (65  to  70  per 
cent  for  river  sand)  gravity  is  competent  to 
transport  without  the  aid  of  current,  and  the 
stream's  capacity  is  infinite.  All  these  factors 
depend  on  slope,  and  as  the  increment  to 
capacity  verges  on  infinity  in  approaching  the 


-us 


+  8 


(llaj 


/tea) 


.1  .2  .3  .4  .5  .6  .7  .8  .9 

Slope 

FIGURE  21. — Extrapolated  curves  of  C=/(S)  for  tentative  equations  of  interpolation  and  for  slopes  less  than  0.8  per  cent. 


slope  which  limits  variation,  it  is  highly  prob- 
able that  capacity  grows  continuously  with 
slope. 

This  criterion  suffices  for  the  rejection  not 
only  of  the  specific  formulas  (6a)  and  (lla), 
but  also  of  their  types,  (6)  and  (11).  In  for- 
mula (6a)  the  occurrence  of  the  maximum 
value  of  C  is  determined  by  the  negative  coeffi- 
cient of  S3;  and  it  is  true  as  a  general  fact  that 
equations  of  the  class  indicated  by  (12)  yield 
maxima  whenever  the  coefficient  of  the  highest 
power  of  the  independent  variable  is  negative. 
It  is  possible,  or  perhaps  probable,  that  if  each 
series  of  laboratory  values  were  to  be  formu- 
lated under  (7)  or  (8)  the  conditions  for  maxima 
would  be  found  to  occur.  On  the  whole,  the 


extrapolations  for  higher  slopes  tend  to  restrict 
choice  to  forms  (9)  and  (10),  with  some  reser- 
vation as  to  forms  (7)  and  (8). 

Figure  21  gives  extrapolated  curves  for 
slopes  less  than  0.8  per  cent  and  represents  the 
same  equations  as  figure  20,  except  that  the 
curve  for  (6a)  is  omitted.  It  will  be  observed 
that  it  magnifies  greatly  the  space  between  0 
and  B  in  figure  20,  the  scale  of  slopes  being  10 
times  and  the  scale  of  capacities  100  times  as 
large.  The  implications  of  the  functions  for 
low  slopes  are  specially  important  because 
extrapolation  from  laboratory  conditions  to 
those  of  natural  streams  will  nearly  always 
involve  the  passage  from  higher  to  lower 
slopes. 


64 


TBANSPOBTATION   OF   DEBRIS   BY   BUNNING   WATER. 


Curves  (7a)  and  (11  a)  reach  the  origin  of 
coordinates — that  is,  their  equations  indicate 
that  at  the  zero  of  slope  there  is  no  capacity 
for  traction.  Formula  (11  a)  gives  small  but 
finite  capacities  for  very  low  slopes;  but  under 
formula  (7a)  finite  capacities  cease  when  the 
slope  falls  to  0.29  per  cent,  and  for  lower  slopes 
there  is  indication  of  negative  capacities.  If 
the  conditions  of  traction  permitted,  negative 
capacity  might  be  interpreted  as  capacity  for 
traction  upstream ;  but  as  this  is  inadmissible, 
the  negative  values  may  be  classed  as  surd 
results  arising  from  the  imperfection  of  the 
correlation  between  an  abstract  formula  and  a 
concrete  problem.  The  curves  of  (8a)  and  (9a) 
also  intersect  the  axis  of  slope  at  some  distance 
from  the  origin,  and  their  extensions  indicate 
negative  capacity.  The  curve  of  (lOa)  becomes 
tangent  to  the  axis  of  slope  at  the  point  corre- 
sponding to  a  slope  of  0.39  per  cent  and  there 
ends,  having  no  continuation  below  the  axis. 
It  is  the  real  limb  of  a  parabola  of  which  all 
other  parts  are  imaginary.  It  expresses  to  the 
eye  the  implication  of  formula  (lOa)  that  trac- 
tion ceases  when  the  slope  is  reduced  to  0.39 
per  cent,  and  that  its  cessation  is  not  abrupt 
but  gradual;  and  also  the  implication  of  the 
general  formula  (10)  that  traction  ceases  when 
the  slope  is  reduced  to  the  value  a. 

It  is  a  matter  of  observation  that  when  slope 
is  gradually  reduced,  the  current  becoming 
feebler  and  the  capacity  gradually  less,  the 
zero  of  capacity  is  reached  before  the  zero  of 
slope.  For  each  group  of  conditions  (fineness, 
width,  discharge)  there  is  a  particular  slope 
corresponding  to  the  zero  of  capacity.  It  is 
also  a  matter  of  observation  that  the  change 
in  capacity  near  the  zero  is  gradual.  Formulas 
(7a),  (8a),  (9a),  and  (lOa)  therefore  accord 
with  the  data  of  observation  in  the  fact  that 
they  connect  the  zero  of  capacity  with  a  finite 
slope;  formula  (Ha),  which  connects  zero 
capacity  with  zero  slope,  is  discordant.  Also, 
formulas  (lOa)  and  (lla)  accord  with  the  data 
of  observation  in  that  they  make  the  approach 
of  capacity  to  its  zero  gradual;  while  formulas 
(7a),  (8a),  and  (9a),  which  make  the  arrival  of 
capacity  at  its  zero  abrupt,  are  in  that  respect 
discordant. 

But  one  of  the  formulas  (lOa),  shows  quali- 
tative agreement  with  both  of  the  criteria 
applied  through  extrapolation  to  low  slopes; 
and  that  formula  is  one  of  the  two  which 


respond  best  to  the  criterion  applied  through 
extrapolation  to  high  slopes.  That  type  of 
formula,  or 

£=&,(£-»)"---  ..(10) 


was  therefore  selected  for  the  reduction  of  the 
more  or  less  irregular  series  of  observational 
values  of  capacity  to  orderly  series  better 
suited  for  comparative  study. 

In  rewriting  the  formula  the  coefficient  is 
changed  from  6  to  6U  because  corresponding 
coefficients  62,  63,  etc.,  are  to  be  used  in  a  series 
of  formulas  expressing  the  relations  of  capacity 
to  various  conditions.  As  slope  is  a  ratio 
between  lengths,  (S  —  a)n  is  of  zero  dimensions 
and  &!  is  of  the  unit  C;  it  is  the  value  of  capacity 
when  S  —  a=l. 

The  slope  which  is  barely  sufficient  to 
initiate  traction  has  been  defined  (p.  35)  as 
the  competent  slope.  To  whatever  extent  a 
represents  the  competent  slope  the  formula 
has  a  rational  basis.  The  local  potential 
energy  of  a  stream,  or  the  energy  available  at 
any  cross  section  in  a  unit  of  time,  is  simply 
proportional  to  the  product  of  discharge  by 
slope  or,  if  the  discharge  be  constant,  is  pro- 
portional to  the  slope.  So  long  as  the  slope 
is  less  than  that  of  competence  the  energy  is 
expended  on  resistances  at  contact  with  wetted 
perimeter  and  air  and  on  internal  work 
occasioned  by  those  resistances.  When  the 
slope  exceeds  the  competent  slope,  part  of  the 
energy  is  used  as  before  and  part  is  used  in 
traction.  The  change  from  competent  slope 
to  a  steeper  slope  increases  the  available 
energy  by  an  amount  proportional  to  the 
increase  of  slope,  and  the  increase  of  energy  is 
associated  with  the  added  work  of  traction. 
Capacity  for  traction,  beginning  at  competent 
slope,  increases  pari  passu  with  the  increase 
of  the  excess  of  slope  above  the  competent 
slope,  and  there  is  manifest  propriety  in  treat- 
ing it  as  a  function  of  the  excess  of  slope 
rather  than  of  the  total  slope.  It  is  of  course 
also  a  function  of  the  total  slope;  but  an 
adequate  formula  for  its  relation  to  the  excess 
of  slope  may  reasonably  be  supposed  to  be 
simpler  than  a  formula  for  its  relation  to  total 
slope.  If  a  represents  competent  slope,  then 
the  relation  of  capacity  to  S  —  a  should  be 
simpler  than  its  relation  to  8. 

Instructive  information  as  to   the  relative 
simplicity  of  the  two  functions  is  obtained  by 


ADJUSTMENT   OF   OBSERVATIONS. 


65 


comparing  their  logarithmic  graphs.  In  figure 
22  the  curved  line  AB  has  been  copied  from  the 
curve  in  figure  19.  It  is  the  graph  of  log  C= 
f  (log  S)  for  grade  (G),  width  0.66  foot,  and  dis- 
charge 0.734  ft.'/sec.  If  for  S  we  substitute 
S  —  0.3,  we  modify  the  graph  by  moving  each 
point  of  it  to  the  left  by  an  amount  equal  to 
log  S  —  log  (S  —  0.3);  and  we  produce  the  lino 
CD,  which  is  the  graph  of  log  C=fa(\og 
(S  —  0.3)).  If  in  similar  manner  we  derive 
the  graph  of  log  C=fm(\og  (S-0.6)),  the  re- 
sult is  the  line  EF.  CD  curves  in  the  same 
direction  as  AB  but  less  strongly;  EF  curves 
in  the  opposite  direction.  It  is  evident  that 


the  three  curves  belong  to  a  continuous  series, 
and  that  somewhere  between  CD  and  EF  a 
member  of  the  series  is  straight  or  approxi- 
mately straight.  That  straight  line,  GH,  is 
the  graph  of  log  C=fIVQog  (5-0.39));  but  as 
it  is  straight,  its  equation  may  be  written 

log  <7=log  6,  +  n-log  GS-0.39), 

in  which  log  6t  is  the  ordinate  of  the  inter- 
section of  the  line  with  the  axis  of  log  C,  and  n 
is  the  trigonometric  tangent  measuring  the 
inclination  of  the  line  to  the  axis  of  log  S. 
This  is  identical  with  equation  (15)  except 
that  0.39  appears  in  place  of  a;  and  in  fact 


200 


100 
80 


00 


20 


7 


.4  £ 

Slop* 


.8        1.0 


Z.Q 


FIGCBE  22.— The  relation  of » in  C=f(S-r)  to  the  curvature  of  the  logarithmic  graph. 


the  value  of  a  in  equation  (lOa)  was  computed 
graphically  by  means  of  the  logarithmic  plot. 

The  line  AB,  being  the  graph  of  log  C— 
/.Gog  S),  is  also  the  logarithmic  graph  of 
C=f(S).  The  line  GE,  being  the  graph  of 
log  C^/jvGog  (8  —  a))  is  also  the  logarithmic 
graph  of  <7=/v(S  —  a).  Their  relation  in  re- 
spect to  simplicity  is  that  of  the  curve  to  the 
straight  line. 

In  view  of  these  suggestions  of  harmony  it  is 
peculiarly  pertinent  to  inquire  whether  a  is 
actually  representative  of  competent  slope: 
and  it  will  be  convenient  to  make  that  inquiry 
in  connection  with  the  determination  of  values 

20921°— No.  86—14 5 


of  a  for  the  several  series  of  observations  on 
capacity  and  slope. 

THE  CONSTANT  a  AND  COMPETENT  SLOPE. 

In  the  experimental  data  for  graded  debris — 
Table  4,  (A)  to  (H) —  are  117  series  of  values 
of  capacity  and  slope.  After  these  had  been 
plotted  and  inspected  in  a  comparative  way, 
it  was  decided  to  restrict  the  main  discussions 
to  92  series  only,  the  discarded  series  being  all 
short  as  well  as  somewhat  discrepant  among 
themselves.  Of  the  92  series  retained,  only 
30  afford  information  as  to  the  correspond- 
ing values  of  o;  that  is,  only  30  of  the  loga- 


66 


TRANSPORTATION    OF    DEBRIS    BY    RUNNING    WATER. 


rithmic  plots  exhibit  curvature  so  definitely 
that  the  approximate  magnitudes  of  the  con- 
stants necessary  to  eliminate  it  can  be  inferred. 
In  some  of  the  remaining  series  the  observa- 
tions are  not  so  distributed  as  to  bring  out  the 
curvature.  For  others  the  observational  posi- 
tions on  the  plots  are  too  -widely  dispersed  to 
give  good  indication  of  the  character  of  the 
best  representative  line. 


On  the  30  logarithmic  plots  the  curves  ap- 
proximately representing  the  observations  were 
drawn,  and  the  values  of  a  necessary  to  replace 
the  curves  by  straight  lines  were  computed 
graphically  in  the  manner  just  indicated. 
These  values  are  given,  to  the  nearest  tenth  of 
1  per  cent  of  slope,  in  Table  7,  where  they  are 
arranged  with  reference  to  the  conditions  of 
the  experiments. 


TABLE  7. —  Values  of  a  in  C=bl(S—a)n,  estimated  from  logarithmic  plots  of  observations. 


Grade 

F 

Range 
of  F 

W 
(feet). 

Values  otffoi  discharge  (ft.3/sec.)  of— 

0.093       0.182 

0.363 

0.545 

0.734 

0.923 

1.021    ;    1.119 

(A) 
(B) 

(C) 

(D) 
(E) 
(F) 
(G) 
(H) 

25,200 
13,400 

5,460 

1,460 
142 
22.1 
5.9 
2.0 

1.77 
2.35 

2.85 

3.58 
12.65 
2.15 
2.92 
2.51 

0.66 
1.00 
1.32 
1.96 

.23 
.44 
.66 
1.00 
1.32 
1.96 

.44 
.66 
1.00 
1.32 
t.96 

.66 
1.00 
1.32 

.66 
1.00 
1.32 

.66 
1.00 
1.32 

.66 
1.00 
1.32 

.66 

- 

- 

- 

0.2 

0.1 

0.7 
.5 

.4 
.0 

i 

.1 

— 

0.1 

.4 
.2 

.1 
.3 

.1 
.1 
3 

0.1 

.1 
.1 

— 

•  — 

- 

— 

- 

— 

— 

- 

.0 

.0 

0 

0.3 
.3 

.4 

!    .; 

!          .6 

.3 



.1 

fi 

NOTE.— The  horizontal  dashes  indicate  series  of  observations  to  which  values  of  <r  are  to  be  assigned. 


The  same  table  shows  the  distribution  of  the 
experimental  series  which  fail  to  give  values  of 
a  but  to  which  it  is  proposed  to  assign  values. 
In  order  to  assign  these  values  properly  it  is 
necessary  to  know  the  laws  of  variation  of  a 
with  reference  to  the  conditions  of  experimen- 
tation. These  are  suggested  in  part  by  the 
roughly  determined  values  of  the  table  and 
are  otherwise  indicated  by  general  considera- 
tions. The  variations  are  connected  with  at 
least  three  conditions — discharge,  width,  and 
fineness — and  are  less  surely  connected  with 
range  of  fineness. 

Considering  first  the  variation  of  a  with  dis- 
charge and  giving  attention  in  the  table  to 
values  of  a  falling  in  the  same  horizontal  line, 
we  find  by  inspection  that  invariably  the  value 


for  a  larger  discharge  is  either  less  than  or 
equal  to  the  value  for  the  corresponding  smaller 
discharge.  The  indication  is  that  a  is  a  de- 
creasing function  of  discharge.  This  relation 
might  have  been  inferred  from  general  con- 
siderations, on  the  theory  that  a  represents 
competent  slope.  Competent  slope  is  the  slope 
giving  competent  velocity  along  the  bed,  but 
bed  velocity  also  varies  directly  with  discharge. 
With  large  discharge  less  slope  is  necessary  to 
induce  competent  velocity;  with  small  dis- 
charge more  slope.  In  other  words,  compe- 
tent slope  varies  inversely  with  discharge.  In 
a  preliminary  discussion  of  the  traction  data 
for  debris  of  grades  (B)  and  (C)  it  was  found 
that  capacity  is  more  sensitive  to  changes  in 
slope  than  to  changes  in  discharge.  The  rela- 


ADJUSTMENT    OF    OBSERVATIONS. 


67 


tion  of  capacity  to  slope  and  discharge  jointly 
being  tentatively  represented  by 

C  cc  SnQm (16) 

the  values  of  n  and  m  were  computed  for  many 
different  conditions,  and  it  was  found  that  on 

atv\ 

the  average  —  =  0.34.     There  is,  however,  con- 

71- 

siderable  variation  in  the  ratio,  and  it  is  rela- 
tively large  when  C  is  small.  On  plotting  the 
values  of  the  ratio  in  relation  to  C  and  graphic- 
ally extrapolating,  it  was  found  that  when 
(7=0  the  ratio  is  about  0.5.  The  condition  of 
zero  capacity  is  that  of  competent  bed  velocity 
and  competent  slope. 

For  any  particular  value  of  C  in   (16)  the 
product  of  Sn  by  Qm  is  constant,  or 


whence 


But,  as  we  have  just  seen,  when  <7has  the  par- 

M 

ticular  value  0=0,  —  =  0.5.  Therefore  compe- 
tent slope  varies  inversely  as  Q°-5.  Its  assumed 
representative,  a,  is  assumed  to  vary  inversely 
with  the  square  root  of  discharge.1 

Turning  now  to  the  relation  of  a  to  width, 
w,  and  examining  Table  7,  we  see  that  a  is  not 
exclusively  either  an  increasing  or  a  decreas- 
ing function  of  w.  Where  the  smallest  widths 
are  concerned,  as  with  Q  =  0.093  ft.3/sec.  and 
Q  =  0.182  ft.3/sec.,  for  grades  (B)  and  (C),  the 
function  is  decreasing.  Where  the  greater 
widths  are  concerned  it  is  for  the  most  part  in- 
creasing. There  is  a  rational  explanation  for 
such  double  relationship  in  the  case  of  compe- 
tent slope. 

Figure  23  represents  two  troughs  in  cross 
section.  Each  has  a  bed  of  debris,  and  they 
are  assumed  to  be  carrying  the  same  total 
discharge.  In  the  wider  there  is  less  discharge 
for  each  unit  of  width,  and  the  tendency  of  the 
smaller  discharge  is  to  reduce  velocity.  There- 
fore to  maintain  a  particular  velocity — namely, 

i  The  preliminary  discussion  on  which  is  based  a  cc  -^-.  did  not  have 

y«.8 

the  advantage  of  the  formula  using  9.  A  rediscussion,  to  be  found  in 
Chapter  V,  yields  a  oc  gj^t  but  it  was  not  practicable  to  give  the  adjust- 

ment  the  benefit  of  this  later  work  without  repeating  the  greater  part 
of  the  computations  of  the  paper.  It  is  not  believed  that  the  advantage 
to  the  results  would  be  commensurate  with  the  labor  involved. 


the  competent  velocity — the  tendency  is  to 
produce  a  relatively  steep  slope.  So  far  as 
this  factor  is  concerned,  competent  slope  is 
relatively  steep  for  a  wider  trough.  But  there 
is  another  factor;  the  velocity  is  influenced  by 
the  resistance  of  the  sides  of  the  trough.  This 
resistance  is  greater  where  the  water  is  deeper, 
because  the  surface  of  contact  is  broader;  and 
the  water  is  deeper  in  the  narrower  trough. 
The  tendency  of  the  resistance  is  to  reduce 
velocities  and  therefore  to  make  the  slope  for 
competent  velocity  steeper  in  the  narrower 
trough. 


Water 


FIGURE  23. — Diagrammatic  sections  of  laboratory  troughs,  illustrating 
relation  of  current  depth  to  trough  width. 

In  very  wide  troughs  the  influence  of  the 
sides  is  of  minor  importance,  the  influence  of 
discharge  per  unit  width  dominates,  and  the 
competent  slope  varies  directly  with  the  width. 
In  very  narrow  troughs  the  influence  of  the 
sides  dominates,  and  the  competent  slope 
varies  inversely  with  the  width.  For  some 
intermediate  width  the  two  tendencies  are 
balanced,  and  the  competent  slope  has  its 
minimum  value.  In  figure  24  abscissas 
measured  from  0  represent  width  of  trough, 
and  ordinates  represent  competent  slope.  The 
curve  sketched,  while  not  quantitative,  has 
adequate  basis  for  its  broader  features  and 


FIGURE  24. — Ideal  curve  of  competent  slope  ( C.  S.)  in  relation  to  width 
of  trough  (W). 

shows  the  general  character  of  the  relation  of 
competent  slope  to  width.  From  its  minimum 
it  ascends  gradually  on  the  side  of  greater 
width,  and  on  the  side  of  lesser  width  rises  with 
relative  rapidity  toward  a  vertical  asymptote 
near  the  axis  of  competent  slope.  These 
various  characters  are  approximately  paralleled 
by  the  variations  of  a  as  shown  in  the  table. 

The  relation  of  a  to  fineness  does  not  come 
out  very  clearly  in  Table  7,  and  a  different 


68 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


arrangement  is  therefore  given  in  Table  8, 
which  omits  the  values  of  a  not  affording  com- 
parison and  adds  three  interpolated  values. 
The  general  fact  thus  shown  is  that  a  increases 
as  fineness  diminishes,  but  to  this  there  are  two 
exceptions.  The  more  important  exception 
is  in  grade  (E),  of  which  all  values  of  a  are  less 
than  the  corresponding  values  for  either  grade 
(C)  or  grade  (G).  The  values  for  grades  (B) 
and  (C)  appear  to  be  about  the  same,  although 
the  two  grades  differ  notably  in  fineness.  The 
exceptions  are  associated  with  peculiarities  of 
the  grades  with  respect  to  range  of  fineness. 
In  grade  (E)  the  range  for  bulk  fineness  is  about 
twice  as  great  as  in  any  of  the  other  grades; 
and  if  this  character  indicates  the  cause  of  its 
abnormally  small  values  of  a,  then  the  abnor- 
mally high  values  in  grade  (B)  are  explained 
by  its  small  range  of  fineness  as  compared  to 
the  range  of  grade  (C). 

TABLE  8. —  Values  of  a,  from  Table  7,  arranged  to  show  vari- 
ation in  relation  to  fineness  of  debris. 


Grade        

(B).        (C). 

(E). 

(0). 

(H). 

Fineness  (  /•'j)  

13,400 

5,460 

142 

5.9 

2.0 

Range  of  fineness.  . 

2.35 

2.85 

12.65 

2.92 

2.51 

Width. 

Discharge. 

Values  of*. 

0.44 
.44 
.66 
.66 
1.00 
1.00 
1.00 
1.32 

0.093 
.182 
.363 
.734 
.182 
.363 
.734 
.734 

0.5 
.0 

0.4 

0.6 
.4 

"'6."6' 

.r 

.1 
.1 

.  i 

0.0 
.0 
.0 

.1 
.1 

.6 
.3 
.4 

! 

The  values  of  a  inferred  from  the  logarithmic 
plots  are  not  sufficiently  precise  to  yield 
quantitative  laws  of  the  relations  of  a  to 
fineness  and  the  range  of  fineness,  and  it  is 
again  necessary  to  seek  instead  the  laws 
governing  competent  slope.  Does  competent 
slope  vary  inversely  with  fineness,  and  what 
is  its  law?  Does  competent  slope  vary  in- 
versely with  range  of  fineness,  and  what  is  its 
law? 

Inasmuch  as  fine  debris  is  moved  by  a  rela- 
tively slow  current  and  as  the  force  of  the  cur- 
rent is  a  direct  function  of  slope,  the  competent 
slope  for  fine  debris  is  less  than  for  coarse.  If 
we  accept  the  thesis  of  Leslie  and  Hopkins  (see 
p.  16)  that  competent  bed  velocity  (V^)  varies 
with  the  sixth  root  of  the  volume  or  mass  of 


the  debris  particle,  then,  since  bulk  fineness  is 
the  reciprocal  of  volume, 

T  ..A  oc  - 


If  we  assume,  from  the  Chezy  formula,  that 
mean  velocity  of  current  is  proportional  to  the 
square  root  of  slope  and  apply  it  to  component 
mean  velocity  (Vcm)  and  competent  slope 
(Sc),  we  have 

I"      oc  fl  °-5 
'  cm   "t  »« 

If  we  further  assume  that  bed  velocity  is  pro- 
portional to  mean  velocity,  then,  by  combining 
the  three  proportions  and  reducing,  we  obtain 


oc 


I 

F2°-33" 


(17) 


As  each  of  the  three  assumed  laws  is  subject  to 
important  qualifications,  the  product  of  their 
combination  must  be  regarded  as  but  a  rough 
approximation  to  the  law  connecting  com- 
petent slope  with  fineness. 

Further  light  on  the  law  is  afforded  by  some 
experiments  made  for  the  specific  purpose  of 
determining  competent  slope.  In  these  experi- 
ments the  discharge  remained  constant  while  the. 
velocity  was  modified  by  changing  the  width  of 
the  outfall.  The  slope  of  the  bed  had  been  pre- 
pared in  advance,  and  the  slope  of  the  water 
surface  was  measured  for  each  width  of  out- 
fall. At  each  stage  of  the  experiment  the 
movement  of  grains  of  debris  along  the  bottom 
was  noted  by  such  phrases  as  "many,"  "sev- 
eral," "few,"  "very  few,"  "none,"  the  words 
being  used  in  that  order  as  a  sort  of  scale. 
Competent  slope  was  inferred  from  a  compari- 
son of  these  notes  with  the  recorded  slopes  of 
the  water  surface.  The  results  are  given  in 
Table  9. 

In  a  closely  related  series  of  experiments, 
Table  10,  a  slope  of  debris  was  prepared  in 
advance,  a  small  discharge  was  passed  over  it, 
and  the  discharge  was  progressively  increased, 
with  notes  on  the  movement  of  debris  grains. 
These  experiments  gave  competent  discharge. 

In  each  series  the  depths  were  measured,  and 
from  these  the  mean  velocities  (Vm)  were 
computed. 

The  experimental  determinations  were  indefi- 
nite for  several  reasons.  In  the  first  place, 


ADJUSTMENT   OF   OBSERVATIONS. 


69 


a  stream  with  the  critical  bottom  velocity  does 
not  transport  debris  and  therefore  does  not 
establish  its  own  slope  of  bed.  The  slopes 
artificially  prepared  were  of  necessity  imper- 
fectly adjusted.  In  the  next  place,  the  pre- 
pared slopes  did  not  imitate  the  natural 
diversity  of  detail  but  were  plane.  When  a 
current  of  competent  velocity  passed  over  one 
of  them  it  immediately  began  to  shape  the  bed 
into  dunes,  and  as  the  modeling  proceeded  the 
activity  of  transportation  increased.  After  the 
dunes  were  formed,  a  smaller  general  velocity, 
or  a  less  discharge,  or  a  lower  general  slope 
was  competent.  The  experiments  being  made 


in  sets,  the  first  of  a  set  gave  a  result  from  the 
plane  bed  and  the  others  from  a  more  or  less 
diversified  bed. 

In  the  third  place,  the  particles  composing 
one  of  the  experimental  grades  of  debris  were 
not  of  uniform  mobility.  Not  only  were  they 
of  diverse  size,  as  indicated  by  the  "range  of 
fineness,"  but  they  were  different  in  shape  and 
in  specific  gravity,  so  that  some  were  able  to 
resist  a  considerably  stronger  current  than 
others.  The  competent  slope  for  the  least 
mobile  particles  was  materially  steeper  than 
that  for  the  most  mobile,  and  no  mode  of 
gaging  average  mobility  was  discovered. 


TABLE  9. — Experimental  data  on  competent  slope. 


Grade 
of 
debris. 

Width 
of 
trough. 

Width 
of  con- 
tractor. 

Dis- 
charge. 

Slope 
of 
debris. 

Slope 
of 
water. 

Depth. 

Mean 
veloc- 
ity. 

• 
Notes  on  movement  of  debris. 

Feel. 

Feet. 

Ft.  '/see. 

Per  ct. 

Per  ct. 

Feet. 

Ft./sec. 

(B)  

1.00 

0.30 
.40 

0.363 
.363 

0.048 
.03 

0.01 
.06 

0.490 
.425 

0.74 
.85 

A  very  few  grains  moving.    Small  dunes  forming. 
Many  grains  moving.    Dunes  forming. 

.30 

.363 

.028 

.03 

.475 

.76 

A  very  few  grains  moving.    Some  dunes  forming. 

C)  

1.00 

.30 
.40 

.363 
.363 

.10 
.10 

.01 
.042 

.451 
.391 

.80 
.93 

A  very  few  grains  moving  in  a  few  places. 
Many  grains  moving.    Dunes  forming. 

.30 

.734 

.10 

.045 

Dunes  forming  rapidly. 

(D)  

1.00 

.30 

.363 

.02 

.21 

.465 

.78 

No  grains  moving. 

.40 

.363 

.02 

.34 

.418 

.87 

An  occasional  grain  moving. 

.30 

.40 
.40 

.734 
.734 
.363 

.02 
.02 
.06 

.33 
.57 
.03 

.743 
.658 
.392 

.99 
1.12 
.93 

Very  few,  if  any,  grains  moving. 
Many  grains  moving.    Dunes  forming. 
A  very  few  grains  moving.    No  dunes  forming. 

.40 

.363 

.10 

.055 

Very  few  grains  moving. 

.45 

.363 

.10 

.054 

.373 

.97 

Some  small  grains  moving. 

.50 

.363 

.10 

.056 

.357 

1.02 

Several  small  grains  moving.    No  dunes  forming. 

.55 

.363 

.10 

Many  grains  moving. 

(E)  

1.00 

.40 

.363 

.05 

.043 

.387 

.94 

No  transportation. 

.40 

.734 

.035 

.025 

.618 

1.19 

Do. 

.50 

.363 

.035 

.057 

.357 

1.02 

Do. 

.50 
.60 

.734 
.363 

.035 
.05 

.069 
.086 

.562 
.330 

1.31 
1.10 

Some  transportation  in  lower  half  of  slope. 
Very  few  grains  moving  in  lower  half  of  slope. 

.60 

.363 

.10 

.0% 

.314 

1.16 

A  few  grains  moving. 

(G)  

1.00 

.70 
.87 

.363 
.363 

.40 
.40 

.22 
.29 

.256 

.238 

.42 
.52 

No  grains  moving. 

.87 
.87 
.87 

.363 
.734 
.363 

.50 
.50 
.65 

.46 
.46 
.70 

.225 
.360 
.201 

.61 
.01 
.80 

A  few  grains  moving  at  one  place. 
A  few  grains  moving  in  places. 
Very  few,  if  any,  grains  moving. 

.87 

.363 

.80 

.80 

.184 

.98 

An  occasional  gram  moving. 

.87 

.363 

1.00 

.93 

.173 

2.10 

Several  grains  moving  in  center  half  of  slope. 

When  an  experiment  was  begun  with  a  velocity 
well  below  competence,  and  the  velocity  was 
gradually  increased,  the  first  movement  de- 
tected would  be  the  saltation  of  some  small  or 
light  particle,  and  then  the  number  of  particles 
moving  would  gradually  grow  with  the  quicken- 
ing of  current. 

An  attempt  to  correlate  the  "  notes  on  move- 
ment of  de'bris"  in  Table  10  for  a  discharge  of 
0.363  ft.3/sec.  gave  the  following  values  of 
water  slope  for  equivalent  phases  of  movement: 

Grade (B)        (C)        (D)        (E)        (G) 

Slope  of  water  (per 
cent) 0.03      0.03      0.06      0.10      0.93 

By  assuming  the  power  -function  Sc  =  a  F2",  or 
log  Sc  =  log  a  +  n  log  F2,  and  plotting  log  Sc  in 


relation  to  log  F2,  the  value  found  for  n  is  about 
—  0.5,   but  it  has   a  large  uncertainty.     The 


•oo 

S-i 


01234 
LogF 

FIGURE  25.— Logarithmic  plot  of  competent  slope  in  relation  to  fineness 
ofdeVis. 

plot,  figure  25,  shows  the  competent  slope  for 
grade  (E)  smaller  than  would  be  indicated  by 


70 


TRANSPORTATION    OF    DEBRIS    BY    RUNNING    WATER. 


its  neighbors,  but  not  so  much  smaller  as  is  the 
value  of  a  for  grade  (E)  in  Table  7.  It  also 
makes  the  competent  slope  for  grade  (B) 
aberrant  in  the  same  sense  as  the  corresponding 


value  of  a.  The  data  are  too  vague  to  give 
much  value  to  such  correspondences;  but  the 
broader  resemblances  aid  in  connecting  a  with 
competent  slope. 


TABLE  10. — Experimental  data  on  competent  discharge. 


Grade 
of 
debris. 

Width 
of 

trough. 

Width 
of  con- 
tractor. 

Dis- 
charge. 

Slope 
of 
de'bris. 

Depth. 

Mean 
veloc- 
ity. 

Notes  on  movement  of  debris. 

Feet. 

Feet. 

Flffsec. 

Per  ct. 

Feet. 

Ft.liec. 

(E).... 

1.32 

0.90 

0.039 
.019 

2.00 
1.95 

o.oio 

6.96 

Many  griins  moving.    Dunes  quickly  formed. 
Some  grains  moving;  many  moving  as  small  channels  are  formed  or  surface  becomes 
rough. 

.019 

1.00 

.014 

1.03 

No  grains  moving. 

.039 

1.00 

.026 

1.14 

Many  grains  moving. 

.039 

1.00 

.024 

1.23 

Several  grains  miving.    Dunes  forming. 

.019 

1.00 

.018 

.80 

No  groins  moving. 

1.00 

.72 

.010 

2.00 

.015 

.67 

A  few  grains  moving  in  a  few  places.    Dunes  forming. 

.019 

1.94 

.025 

.76 

Many  grains  moving.    Dunes  forming. 

.92 

.019 

1.00 

A  few  grains  moving  in  a  few  places. 

.029 

1.00 

Several  grains  moving  in  several  places. 

.039 

.50 

.053 

.74 

No  grains  moving. 

.058 

.50 

.060 

.96 

A  few  grains  moving  in  a  few  places. 

.075 

.50 

.065 

1.15 

Severalgrains  moving  in  several  places. 

.093 

.50 

.070 

1.33 

Transportation. 

.60 

.46 

.010 

2.00 

.022 

.69 

A  few  grains  moving  in  a  few  places. 

.019 

2.00 

.028 

1.03 

Several  erains  moving  in  nearly  all  parts  of  trough. 

.010 

1  13 

No  motion. 

.019 

1.13 

.042 

.69 

A  lew  grains  moving  in  a  few  places. 

.039 

1.13 

.050 

1.19 

Many  grains  moving.    Surface  becoming  rough. 

(F)  

.66 

.46 

.039 

2.51 

.043 

1.38 

An  occasional  grain  moving.    Several  groins  moving  after  a  time. 

.058 

1  05 

No  grains  moving. 

.075 
.093 

1.05 
1.05 

.071 
.092 

1.59 
1.52 

A  few  grains  moving  in  a  few  places. 
Several  grains  moving  in  some  places. 

.111 

1.05 

Many  grains  moving. 

1.00 

(?) 

.146 

1.00 

"".'086 

i.70 

Few  grains  moving  in  some  places. 

.164 

1.00 

.093 

1.77 

Several  grains  moving. 

(G)  

.66 

.45 

.146 

1.10 

.119 

1.84 

No  grains  moving. 

.16-1 

.182 

1.10 
1.10 

.126 
.139 

1.95 
1.96 

Very  few  grains  moving. 
A  few  grams  moving  near  middle  of  trough. 

.200 

1.10 

.143 

2.10 

A  few  grains  moving. 

.218 

1.10 

.153 

2.13 

Several  grains  moving  in  lower  half  of  trough. 

.164 
.182 
.200 
.146 

1.68 
1.58 
1.58 
2.05 

.112 
.104 
.104 
.106 

2.20 
2.63 
2.88 
2.07 

Some  grains  moving  in  lower  half  of  trough. 
Several  grains  moving  in  lower  half  of  trough,  but  very  few  in  upper  half. 
Manv  grains  moving  in  lower  half,  but  few  in  upper  half. 
Several  grains  moving  in  middle  of  trough;  many  in  lower  part.    Cutting  of  grade. 

.111 

2.05 

.088 

1.89 

A  few  grains  moving. 

.093 

2.05 

.088 

1.58 

Very  few  grains  moving. 

.075 

2.05 

.070 

1.61 

No  grains  moving. 

.058 

2.50 

.052 

1.61 

Do. 

.075 

2.50 

.054 

2.08 

A  few  grains  moving. 

.093 

2.50 

.072 

1.95 

Many  grains  moving. 

1.00 

.70 

.182 

1.00 

.124 

1.47 

No  grains  moving. 

.218 

1.00 

.141 

1.  55 

No  (or  very  few)  grains  moving. 

.  2aS 

1.00 

.153 

1.67 

Very  few  moving. 

.290 

1.00 

.159 

1.84 

Many  grains  moving. 

.182 

1.45 

.089 

2.04 

Very  few  small  grains  moving. 

.218 

1.45 

.098 

2.23 

Several  grains  moving. 

.255 

1.45 

.105 

2.43 

Many  grains  moving. 

.182 

1.90 

.090 

2.02 

Several  grains  moving. 

.146 

1.90 

.080 

1.83 

Very  few  grains  moving. 

.111 

1.90 

.059 

1.88 

No  grains  moving. 

.093 

2.52 

.061 

1.52 

Do. 

.111 

2.52 

.067 

1.66 

A  very  few  grains  moving  in  part  of  trough. 

.128 

2.52 

.074 

1.73 

A  few  grains  moving,  except  near  head  of  trough. 

.140 

2.52 

Many  grains  moving.    Grade  cutting  in  places. 

1.32 

.92 

.272 

1.05 

"".'iis 

1  79 

No  grains  moving. 

.327 

1.05 

.125 

1.99 

Very  few  grains  moving. 

.345 

1.05 

.128 

2.04 

A  few  grains  moving. 

.363 

1.06 

.130 

2.11 

Several  grains  moving. 

.218 
.237 
.255 

1.40 
1.40 
1.40 

.075 
.079 
.084 

2.21 
2.27 
2.27 

Very  few  grains  moving  near  center  of  trough. 
None  moving  except  at  middle  of  trough. 
Very  few  moving  except  at  middle  of  trough. 

.272 

1.40 

.090 

2.28 

A  few  grains  moving. 

.290 

1.40 

.099 

2.22 

Several  moving  in  middle  and  lower  parts;  very  few  in  upper. 

.218 

2.00 

.079 

2.09 

A  few  grains  moving. 

.237 

2.00 

.090 

1.99 

Several  grains  moving  in  middle  part  of  trough. 

.255 

2.00 

.097 

1.98 

Several  grains  moving. 

.146 

2.50 

.055 

2.01 

Very  few  grains  moving. 

.164 

2.50 

.061 

2.04 

Several  grains  moving. 

.182 

2.50 

.069 

1.99 

Do. 

(H)... 

.66 

.46 

.363 

1.10 

.212 

2.57 

Occasionally  a  grain  moving. 

.454 
.545 

1.10 
1.10 

.260 
.399 

2.62 
2.74 

A  few  grains  moving  in  lower  half  of  trough. 
Several  grains  moving  in  lower  two-thirds  of  trough,  and  occasionally  a  grain  in  upper 
third. 

.45 

.639 
.272 
.308 
.345 

1.10 

1.30 
1.30 
1.30 

.329 
.163 
.171 

.184 

2.91 
2.49 
2.70 
2.82 

Several  grains  moving  in  middle  third  of  trough;  a  few  in  upper  and  lower  parts. 
Occasionally  one  grain  moving. 
Some  grains  moving  below  middle;  1  to  3  grains  in  a  cross  section. 
2  to  5  grains  moving  in  a  cross  section. 

.182 

2.05 

.132 

2.05 

No  grains  moving. 

.218 

2.05 

.137 

2.39 

A  few  grains  moving  in  lower  three-fourths  of  trough. 

.255 

2.05 

.141 

2.72 

Many  grains  moving  in  lower  three-fourths  ol  trough. 

ADJUSTMENT   OP   OBSEBVATIONS. 


71 


Iii  treating  the  data  of  Table  9,  the  assump- 
tion was  made  that  the  note  "several  grains 
moving"  corresponds  to  competent  slope;  and 
averages  were  found  of  the  values  of  mean 
velocity  and  depth.1  These  are: 

Grade (E)         (F)         (G)         (H) 

Mean  velocity  (ft./sec.)..  1.10        1.52        2.14        2.83 
Depth  (foot) 0.020      0.092      0.108      0.218 

When  the  logarithms  of  these  numbers  are 
plotted  the  positions  fall  well  in  line,  and  the 
representative  line  gives  (F^,  indicating  the 
mean  velocity  corresponding  to  competent  bed 
velocity) 

M>:M -  (18) 


Assuming  again  that  bed  velocity  is  propor- 
tional to  mean  velocity,  and  again  assuming 
the  validity  of  the  Chezy  formula,  we  obtain 
from  (18) 

<Se«-*L  -(19) 


The  two  values  of  the  exponent  of  F2  derived 
from  the  experiments,  namely,  —0.44  and 
—  0.50,  are  both  larger  than  the  deductive 
value,  —0.33,  of  equation  (17),  but  the  dispar- 
ity is  quite  natural  in  view  of  the  indefiniteness 
of  the  data  and  the  uncertainties  of  the  assump- 
tions. Collectively  the  values  indicate  an  order 
of  magnitude. 

The  influence  of  range  of  fineness  on  compe- 
tent slope  appears  to  be  of  the  same  nature  as 
its  influence  on  a,  though  much  less  pronounced, 
but  the  determinations  of  competent  slope  are 
too  indefinite  to  give  the  greatest  value  to  the 
comparison.  It  is  significant,  however,  that 
while  the  logarithmic  plots  for  grade  (E)  and 
width  1.00  and  for  three  different  discharges 
(Table  7)  all  yield  values  of  a  less  than  0.05  per 
cent,  the  experiments  on  competent  slope  (Ta- 
ble 9)  record  for  one  of  the  discharges  "no 
grains  moving"  with  a  slope  of  0.21  per  cent, 
and  for  another  "very  few,  if  any,  grams  mov- 
ing" with  a  slope  of  0.33  per  cent.  The  values 
of  a  in  this  case  fall  far  below  those  for  the  most 
mobile  components  of  the  debris  grade  which 
has  the  largest  range  of  fineness.  The  general 
facts  appear  to  be  that  a  varies  decreasingly 

1  A  few  series  of  observations  on  competent  velocity  have  been  made 
by  others.  They  pertain  chiefly  to  flume  traction  and  are  cited  in 
Chapter  XII.  Login,  whose  results  are  given  in  Chapter  VII,  used  the 
methods  of  stream  traction  but  omitted  to  measure  the  sizes  of  materials 
transported. 


with  range  of  fineness,  and  that  competent 
slope  is  subject  to  a  variation  of  the  same  kind, 
which  may  or  may  not  be  of  the  same  magni- 
tude. 

The  cause  of  this  variation  is  not  surely 
known,  but  a  plausible  suggestion  in  regard  to 
it  may  be  made.  In  the  experiments  with 
mixtures  of  two  or  more  grades  it  was  found  that 
before  the  slope  had  been  established,  espe- 
cially when  low  velocities  were  used,  the  cur- 
rent tended  to  sort  the  debris,  building  deposits 
with  the  coarser  part  and  delivering  the  finer 
material  at  the  end  of  the  trough.  In  experi- 
ments with  a  single  grade  the  same  tendency 
doubtless  existed.  It  was  in  fact  observed  in. 
connection  with  dunes  and  antidunes,  which 
sometimes  showed  a  shading  in  color  due  to 
partial  sorting  with  respect  to  density,  the 
heavier  particles  being  dark,  the  lighter  pale. 
As  the  differences  of  size  within  a  grade  were 
not  such  as  to  appeal  strongly  to  the  eye,  con- 
siderable sorting  with  respect  to  size  might 
take  place  without  attracting  attention.  With 
the  ordinary  routine  of  the  experiments,  which 
began  in  each  series  with  low  slopes  and  veloci- 
ties and  gradually  increased  them,  the  influ- 
ences of  such  sorting  may  have  been  systematic, 
and  thus  may  have  modified  that  relation  of 
values  which  finds  expression  in  the  constant  a. 
The  result  of  such  influence  would  be  more  pro- 
nounced for  grade  (E)  than  for  grades  with 
smaller  range  of  fineness.  It  is  easy  to  see  also 
that  an  allied  influence  may  have  affected  to 
some  extent  the  interpretation  of  the  experi- 
ments on  competent  slope. 

The  variations  of  a  are  paralleled  in  so  many 
ways  by  the  variations  of  competent  slope  as 
to  leave  little  doubt  that  the  one  is  in  some 
way  representative  of  the  other.  It  can  not 
be  said  that  the  constant  a,  arbitrarily  intro- 
duced to  rectify  curves  and  thereby  facilitate 
interpolation,  is  the  equivalent  of  the  slope  of 
competence — if  for  no  other  reason  than  that 
the  competent  slope  for  a  grade  of  de'bris  made 
of  unequal  grains  eludes  precise  definition — 
but  it  may  well  be  a  complex  function  of  the 
competent  slopes  of  all  the  different  sorts  of 
grams  contained  in  one  of  the  laboratory 
grades. 

For  the  practical  purpose  of  obtaining  values 
of  a  for  use  in  formulas  of  interpolation,  the 
preceding  discussion  yields  a  large  body  of 
pertinent  information.  Sigma  varies  inversely 


72 


TRANSPORTATION    OF    DEBRIS    BY    RUNNING    WATER. 


with  a  power  of  discharge,  approximately  the 
0.5  power.  It  varies  inversely  with  a  some- 
what smaller  power  of  the  bulk  fineness  of 
de'bris.  It  varies  inversely  with  the  range  of 
fineness  of  the  grades  of  debris,  this  variation 
serving  to  qualify  the  preceding.  It  varies 
with  width  of  channel,  the  variation  including 
a  minimum.  Without  attempting  to  give 
definite  symbolic  expression  to  these  laws  of 
variation,  they  were  applied  to  the  practical 
problem,  and  by  a  series  of  adjustments  the 
skeleton  of  values  of  a  in  Table  8  was  developed 
into  a  system  covering  the  whole  range  of 
experimental  conditions.  That  system  is  pre- 
sented in  Table  1 1 . 

TABLE  11. —  Values  of  a,  in  per  cent  of  slope,  as  adjustedfor 
use  in  interpolation  equations  of  the  form  C=bi  (S — a)n. 


Grade. 

Width 
(feet). 

Values  of<r  for  discharge  (ft.3  /sec.)  of  — 

0.093 

0.182 

0.363 

0.545 

0.734 

0.923 

1.021 

1.119 

(A)  
(B)  

(C)  

(D).  . 

0.66 
1.00 
1.32 
1.96 

.23 
.44 
.66 
1.00 
1.32 
1.96 

.44 
.66 
1.00 
1.32 
1.96 

.66 

0.40 

0.20 

.12 
.17 

0  04 

ao? 

.10 

.17 

0.05 



.07 
.10 



...... 



.70 
.50 
.30 

.40 

.16 

.19 

.60 
.10 
.10 
.12 
.17 

.20 
.11 
.15 
.22 

.14 







.08 
.08 
.12 
.18 

".'os' 

.11 
.16 
.24 

.07 
.06 
.10 
.14 

".'6e' 

.09 
.13 
.20 

.08 

.06 
.08 
.12 

"."oi" 

.07 
.11 
.17 

----- 

:;:::: 

"6."io 

•-•-•• 

:::::: 

.06 

"".ii 

(B)  

1.00 
1.32 

.66 

.17 

.12 
.16 

.12 

10 
.11 

0.08 

.09 

.06 

<F)  
(G)  

1.00 
1.32 

.66 
1.00 
1.32 

.66 



.08 

.06 
.04 



.04 
.03 

.03 
.02 

"".17 
.23 

.27 
.33 
.40 

.48 



.33 
.44 

.21 
.31 



.17 
.22 





.39 

.28 

.50 

.36 

(H)  

1.00 
1.32 

.66 

.58 

.41 

.71 

.50 

.80 

.56 

INTERPOLATION. 

The  values  of  a  having  been  assigned,  the 
data  of  Table  4  were  once  more  plotted  on 
logarithmic  paper,  the  ordinates  again  repre- 
senting load  or  capacity,  and  the  abscissas  rep- 
resenting S  —  a,  or  observed  slope  less  the  con- 
stant slope  a.  As  in  the  preliminary  plotting 
for  inspection,  the  primary  data  were  (1)  the 
estimates  of  load  from  the  quantity  of  de'bris 
delivered  at  the  end  of  the  trough  and  (2)  the 
associated  slopes  of  the  bed  of  debris,  and 
accessory  data  were  added  with  distinctive 


notation.  The  secondary  data  were  used 
chiefly  to  indicate  the  relative  precision  of  the 
primary  data,  the  primary  having  greater 
weight  when  agreeing  more  closely  with  the 
secondary.  The  illustrative  plot,  figure  26, 
shows  only  the  primary  data. 

The  next  step  was  to  draw  through  and 
among  the  observational  points  the  best  rep- 
resentative straight  line.  It  is  a  property  of 
the  logarithmic  plot  that  its  distances  repre- 
sent ratios,  and  the  scale  of  ratios  is  every- 
where the  same.  Similar  errors  of  observa- 
tional positions  are  shown  by  similar  distances 
in  all  parts  of  the  plot,  provided  the  errors  are 
considered  as  fractional  parts  of  the  plotted 
quantities.  Each  observational  point  should 
be  given  the  same  influence  in  determining  the 


•o  *° 
& 


FIGURE  26.— Illustration  of  the  method  used  to  adjust  values  of  capacity, 
in  relation  to  slope,  by  means  of  a  logarithmic  plot  of  observed  values 
of  capacity  in  relation  to  slope  minus  a. 

representative  line,  provided  the  (fractional) 
probable  errors  of  the  observations  are  the 
same.  In  this  case,  however,  the  fractional 
probable  errors  for  low  capacities  and  slopes 
are  much  greater  than  for  high  capacities,  and 
the  best  representative  lines  can  not  be  drawn 
without  consideration  of  weights.  Adjust- 
ment by  the  least-squares  method  was  con- 
sidered and  experiments  were  tried,  but  the 
labor  entailed  by  the  necessity  of  using  weights 
was  not  thought  to  be  warranted  by  the 
quality  of  the  data.  The  following  simpler 
and  less  rigorous  method  was  employed : 

A  group  of  observational  points  correspond- 
ing to  the  highest  slopes  was  selected  by  in- 
spection of  the  plot,  and  its  center  of  gravity 
was  computed  by  a  graphic  method.  In  the 


ADJUSTMENT   OF   OBSERVATIONS. 


73 


case  shown  by  figure  26,  the  group  includes 
six  points  and  the  center  of  gravity  is  the 
point  indicated  by  an  arrow.  The  represen- 
tative line  was  made  to  pass  through  this 
point  and  was  otherwise  adjusted  to  position 
bv  eye  estimate,  with  consideration  of  all  the  ob- 
servational points  and  their  supposed  weights. 
By  this  method  the  observations  of  greatest 
weight  were  enabled  to  fix  one  point  on  the 
line  and  were  also  consulted,  along  with  obser- 
vations of  less  weight,  as  to  its  direction  or 
attitude.  The  method  obviously  left  much  to 
personal  judgment,  but  had  a  rigorous  method 
been  attempted  it  would  have  been  difficult  or 
impossible  to  avoid  the  use  of  nonrigorous 
judgment  in  the  assignment  of  weights. 

The  line  when  drawn  is  a  generalized  ex- 
pression, for  a  single  series  of  observations, 
of  the  relation  of  capacity  to  slope  less  a.  It 
is  the  graphic  equivalent,  or  graph,  of  a  specific 
equation  of  the  form 

log  C=  log  &!  +  n  log  (S-a) .  (20) 

which  is  the  logarithmic  equivalent  of 

<7=61(S-<r)n (10) 

The  inclination  of  the  line  (tan  6)  was  next 
measured,  or  computed,  giving  the  numerical 
value  of  n;  and  its  point  of  intersection  with 
the  axis  of  log  C  was  determined,  giving  the 
value  of  &,.  The  values  of  n,  &,,  and  a  for  the 
92  series  are  shown  in  Table  15. 

The  graph  was  used,  instead  of  the  equiva- 
lent equation,  in  computing  values  of  C  cor- 
responding to  systems  of  values  of  S.  The 
results  are  given  in  Table  12  and  constitute  the 
data  for  further  generalizations.  To  compute 
values  of  C  for  a  particular  observational  se- 
ries, the  value  of  a  for  the  series  was  first  sub- 
tracted from  each  value  of  8  in  the  adopted 
system,  then  each  remainder  (S  —  d)  was  ap- 
plied as  an  argument  to  the  graph  and  the 
corresponding  value  of  C  read  off.  The  range 
of  values  thus  computed  and  tabulated  was 
either  limited  by  the  range  of  observational 
values,  or  else  included  a  moderate  extrapola- 
tion, which  never  exceeded  10  per  cent  of  the 
observational  range. 

PRECISION. 

If  the  position  of  the  straight  line  in  each  of 
the  logarithmic  plots  (fig.  26)  had  been  deter- 
mined by  rigorous  methods,  it  would  be  pos- 
sible to  compute  by  rigorous  methods  the  prob- 


able  errors  of  the  quantities  implied  by  its 
position.  As  only  approximate  methods  were 
employed  in  placing  them,  only  approximate 
measures  of  precision  are  attainable,  and  an 
elaborate  treatment  would  be  unprofitable. 
The  precision. of  the  attitude  of  the  line,  cor- 
responding to  the  quantity  n,  has  not  been  esti- 
mated; but  computation  has  been  made  of  the 
precision  of  its  position  in  the  direction  of  the 
axis  of  log  C;  and  also  of  the  precision  of  the 
observations,  on  the  assumption  that  the  dis- 
tances of  the  observational  points  from  the 
straight  line  represent  errors  of  observation. 
The  precision  of  the  position  of  the  line  in- 
volves the  precision  of  the  adjusted  values  of 
capacity,  and  also  the  precision  of  the  coeffi- 
cient of  the  equation  of  adjustment.  The 
method  of  computation,  given  below,  is  also 
the  method  employed  in  various  other  compu- 
tations of  precision,  the  results  of  which  appear 
in  later  chapters. 

Each  logarithmic  plot,  corresponding  to  an 
observational  series,  was  treated  separately. 
The  distance  of  each  plotted  observational 
point  from  the  representative  straight  line,  in 
a  direction  parallel  to  the  axis  of  log  C,  was 
measured.  This  distance,  interpreted  by  the 
scale  of  the  section  paper,  gives  the  logarithm 
of  the  ratio  between  an  observed  capacity  and 
the  corresponding  adjusted  capacity.  By  using 
a  strip  of  the  section  paper  as  measuring  scale, 
it  was  possible  to  read  the  ratio  directly.  It 
was  also  possible,  without  computation,  and  as 
a  simple  matter  of  reading,  to  subtract  unity 
from  the  ratio  and  multiply  the  remainder  by 
100,  thus  recording  directly  the  residual,  or 
observational  error,  as  a  per  cent  of  the  quan- 
tity measured.  The  facility  of  this  operation 
determined  the  estimation  of  all  errors  in  per- 
centage. 

From  the  residuals  thus  obtained,  probable 
errors  were  computed  by  the  following  approxi- 
mate formulas,  in  which  m  is  the  number  of  the 
residuals  and  [v]  is  the  sum  of  the  residuals, 
irrespective  of  sign: 

Probable  error  of  an  observation  = 


0.845 


[v] 


Probable  error  of  adjusted  capacities 
0.845  -M- 


74 


TRANSPORTATION    OF    DEBRIS   BY   RUNNING    WATER. 


Table  12  contains  the  computed  probable 
errors  for  all  those  series,  82  in  number,  in 
which  the  number  of  observations  is  not  less 
than  4.  For  a  smaller  number  of  observations 
it  was  possible  so  to  frame  adjusting  equations 
as  to  leave  no  residuals;  and  although  this  was 
not  done,  the  propriety  of  applying  the  com- 
putation to  such  cases  was  not  evident. 

The  arithmetical  mean  of  the  82  values  of 
probable  error  for  adjusted  capacities  is  ±2.50 
per  cent.  The  corresponding  mean  for  observed 
capacities  is  ±8. 80  per  cent.  Another  measure 
of  the  precision  of  the  observations,  collectively, 
is  the  arithmetical  mean,  irrespective  of  sign, 
of  all  the  residuals  (966  in  number)  involved 
in  the  82  series.  That  mean  is  11.5  per  cent. 

While  these  estimates  of  precision  are  com- 
puted specifically  for  capacities  or  loads,  the 
sources  of  the  errors  are  not  restricted  to  the 
observations  of  loads  but  include  also  the  obser- 
vations of  slope  and  discharge.  With  use  of 
the  same  diagrams  it  would  be  possible  to  con- 
sider slope  as  a  function  of  capacity  and  com- 
pute the  probable  errors  of  slope  determina- 
tions. The  relations  are  such  that,  within  any 
series,  the  probable  error  of  slope,  considered 
as  a  percentage,  is  less  than  the  probable  error 
of  capacity. 

The  residuals  within  a  series  are,  as  a  rule, 
relatively  great  for  the  lower  slopes;  but  this 
is  true  only  when  the  residuals  are  considered 
as  fractional  parts  of  the  capacity  values  to 
which  they  pertain.  If  the  residuals  be 
measured  in  the  unit  of  capacity,  then  they  are 
relatively  great  for  the  higher  slopes.  The 
gradation  of  precision  in  relation  to  slope  was 
discussed  with  some  care,  and  an  elaborate 
system  of  weights  was  prepared  for  the  ad- 
justed values  of  capacity.  While  these  weights 
were  of  service  in  connection  with  various 
combinations  afterward  made,  it  has  not 
seemed  necessary  to  include  them  in  the 
printed  tables.  It  is  to  be  understood,  how- 
ever, that  the  probable  errors  associated  with 
each  series  in  Table  ]2  apply  to  the  values  of 
capacity  as  a  group,  the  probable  errors  (in 
per  cent)  of  the  smaller  values  being  relatively 
great  and  those  of  the  larger  values  relatively 
small. 

The  residuals  represent  chiefly  the  errors  of 
observation,  but  they  include  also  errors  intro- 
duced in  the  process  of  adjustment.  To  what- 


ever extent  the  formula  of  adjustment  mis- 
represents the  actual  relation  between  the 
variables,  to  whatever  extent  the  assigned 
values  of  a  are  inaccurate,  and  to  whatever 
extent  the  representative  lines  of  the  plot  are 
misplaced,  factors  of  error  are  introduced  which 
tend  to  increase  the  residuals.  The  estimates 
of  probable  error  of  observations  are  therefore 
larger  than  they  would  be  if  the  methods  of 
adjustment  were  perfect. 

The  observations  giving  large  percentage 
residuals  were,  as  a  class,  treated  in  the  graphic 
adjustments  as  of  relatively  low  weight;  but  in 
the  computations  of  probable  error  they  were 
treated  as  of  equal  weight.  The  estimates  of 
probable  error,  and  their  average  values,  are 
larger  than  they  would  be  if  computed  with 
regard  to  weights.  The  influence  of  this  factor 
can  not  be  definitely  evaluated,  but  ratios 
brought  out  in  the  discussion  of  residuals  for 
the  assignment  of  weights  indicate  that  it  has 
some  importance.  It  is  thought  that  the 
average  probable  error  of  the  adjusted  values 
may  be  as  low  as  ±2.0  per  cent,  and  that  of  the 
observations  as  low  as  ±7.0  per  cent. 

On  the  other  hand,  the  computations  take 
account  only  of  the  discrepancies  revealed  by 
comparing  observations  of  the  same  series  and 
do  not  cover  such  discrepancies  as  exist  be- 
tween one  series  and  another.  The  estimates 
of  probable  error  are  smaller  than  they  would 
be  if  both  classes  of  discrepancies  were  included. 
This  matter  receives  further  consideration  in 
Chapters  V  and  VI. 

DUTY. 

The  duty  of  water  traction,  as  defined  by  the 
units  adopted  for  this  paper,  is  the  capacity  in 
grams  per  second  for  each  cubic  foot  per  second 

Q 

of  discharge,  and  its  formula  is    U=-^..    The 

V 

duty  corresponding  to  each  adjusted  capacity 
has  been  computed,  and  the  values  appear  in 
Table  12. 

It  is  some  tunes  desirable  to  treat  duty  as  the 
ratio  which  the  mass  of  the  load,  or  capacity, 
bears  to  the  mass  of  the  carrier.  To  obtain  the 
value  of  duty  as  a  ratio  of  masses  the  corre- 
sponding value  of  U  should  be  divided  by 
28,350,  the  number  of  grams  in  a  cubic  foot  of 
water. 


ADJUSTMENT   OF   OBSERVATIONS. 


75 


It  is  sometimes  desirable  to  treat  duty  as  a 
volume  of  debris,  including  voids — that  is,  as 
the  space  occupied  by  debris  as  a  deposit, 
either  before  or  after  transportation.  In  such 
cases  the  American  engineer  commonly  uses 
the  cubic  yard  as  a  unit  of  volume  and  24  hours 
as  a  unit  of  time.  To  obtain  the  approximate 
duty  in  cubic  yards  of  debris  per  24  hours, 
divide  the  duty  in  gm./sec.  by  14.  The  exact 
value  of  the  appropriate  divisor  depends  on  the 


specific  gravity  of  the  debris  particles  and  on 
the  percentage  of  voids  in  the  debris  aggregate. 

EFFICIENCY. 

The  measure  of  efficiency,  as  defined  on  page 

Q 

36,  is  E  ^770-     The   computed   efficiencies  are 
V* 

tabulated  in  Table  12,  along  with  the  adjusted 
capacities  and  duties. 


TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency. 

it.,  grade  of  debris;  to,  width  of  experiment  trough  in  feet;  Q,  discharge  in  cubic  feet  per  second;  S,  slope  of  channel  bed,  in  per  cent;  C,  capa 
for  stream  traction,  in  grams  per  second;  C/,duty  of  water  for  traction,-^:  £,  efficiency  of  current,-^.] 


Conditions. 

Adjusted  data. 

Probable  error 
(per  cent)  of  — 

Or. 

iff 

Q 

S 

C 

V 

E 

Adjusted 
data. 

Observa- 
tion. 

(A) 

0.66 

0.093 

1.2 
1.4 

23.0 
32.3 

247 
347 

206 

248 

1.6 

42.6 

458 

286 

1.8 

53.5 

576 

320 

2.0 

65.5 

704 

352 

2.2 

78.0 

840 

332 

(A) 

.66 

.182 

.7            18.7 

103 

147 

.8  i          25.0 

137 

171 

.9            32.  0 

176 

1% 

1.0            39.5 

217 

217 

1.2:          56.3 

309 

257 

1.4 

75.0 

412 

294 

1.6 

96.0 

528 

330 

1.8 

119 

654 

363 

(A) 

.66 

.545 

.5 
.6 

49.8 
66.0 

91.4 
121 

183 
202 

.7 

83.0 

152 

217 

.8 

101 

185 

231 

.9 

120 

220 

244 

1.0 

140 

257 

257 

1.2  |        184 

358 

282 

(A) 

1.00 

.182 

.5              8.2 
.6            12.5 

45.1 
68.6 

90.2 
114 

.7            17.6 

95.7 

137 

.8            23.5 

129 

161 

.9            30.0 

165 

183 

1.0  1          37.5 

206 

206 

1.2  '          54.5 

300 

250 

1.4             74.0 

407 

291 

1.6 

97.0 

534 

334 

1.8 

122 

670 

372 

2.0 

149 

819 

410 

2.2 

179 

984 

447 

(A) 

1.00 

.363 

.4 
.5 

17.0 
26.9 

46.8 
74.1 

117 

148 

.6 

38.2 

105 

175 

.7 

51.8 

143 

204 

.8 

66 

182 

228 

.9 

83 

229 

254 

1.0 

100 

276 

276 

1.2 

140 

386 

322 

1.4 

185 

510 

364 

1.6 

235 

648 

405 

1.8 

290 

799 

443 

(A) 

1.00 

.734 

.4 
.5 

42.5 
65 

58.0 
88.6 

145 
177 

.6 

91 

124 

207 

.7 

121 

165 

236 

.8 

154 

210 

262 

.9 

191 

260 

289 

1.0 

231 

315 

315 

1.2 

320 

436 

363 

(A) 

1.32 

.182 

.5 

7.0 

38.5 

77 

1.6 

5.2 

.6 

11.2 

61.6 

103 

.7 

16.3 

89.6 

127 

.8 

22.3 

122 

152 

.9 

29.1 

160 

178 

1.0 

36.8 

202 

202 

1.2 

54.3 

298 

248 

1.4 

74.8 

411 

294 

1.6 

98 

538 

336 

76  TRANSPORTATION    OF    DEBRIS   BY    RUNNING    WATER. 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency— Con. 


Conditions. 

Adjusted  data. 

Probable  error 
(percent)  of— 

Or. 

•    w               Q 

s 

C 

V 

E 

Adjusted 
data. 

Observa- 
tion. 

(A) 

1.32 

0.182 

1.8 

124 

681 

378 

1.6 

5.2 

2.0 

152 

835 

418 

2.2 

183 

1,010 

459 

(A) 

1.32 

.363 

.2 

3.6 

9.9 

49.6 

2.3 

7.6 

.3 

10.5 

28.9 

96.3 

.4 

19.5 

53.7 

134 

.5 

30.2 

83.2 

166 

.6 

42.6 

117 

195 

.7 

56.0 

154 

220 

.8 

71 

196 

245 

.9 

87 

240 

267 

1.0 

104 

287 

287 

1.2 

142 

391 

326 

1.4 

183 

504 

360 

1.6 

227 

626 

391 

1.8 

276 

760 

422 

2.0 

328 

904 

452 

(A) 

1.32 

.734 

.3 

27.3 

37.2 

124 

2.3 

7.2 

.4 

48.2 

65.6 

164 

.  5 

73.5 

100 

200 

.6 

103 

140 

233 

.7 

135 

184 

263 

.8 

170 

232 

290 

.9 

219 

285 

317 

1.0 

250 

341 

341 

1.2 

344 

469 

391 

1.4 

445 

606 

433 

(A) 

1.96 

.363 

.4 

12.6 

34.7 

86.8 

2.6 

9.4 

.5 

22.4 

61.7 

123 

.6 

34.0 

93.7 

156 

.7 

47.3 

130 

186 

.8 

72.2 

171 

214 

.9 

79 

218 

242 

1.9 

96 

264 

265 

1.2 

136 

375 

312 

1.4 

178 

490 

350 

1.6 

227 

625 

390 

1.8 

278 

766 

425 

2.0 

335 

923 

462 

(A) 

1.96 

.734 

.2 

11.2 

15.3 

76.5 

3.4 

10.9 

.3 

29.5 

40.2 

134 

.4 

52 

70.8 

177 

.5 

78 

106 

212 

.6 

105 

143 

238 

.7 

136 

185 

264 

.8 

169 

230 

288 

.9 

204 

278 

309 

1.0 

240 

327 

327 

1.2 

320 

436 

363 

(A) 

1.90 

1.119 

.4 

100 

89.4 

224 

3.9 

12.3 

.5 

138 

123 

246 

.6 

180 

161 

268 

.7 

222 

198 

283 

.8 

26« 

238 

298 

.9 

311 

278 

309 

1.0 

359 

321 

321 

(B) 

.23 

.093 

.8 

2.8 

30.1 

37.5 

1.2 

2.6 

.9 

5.3 

57.0 

63.  3 

1.0 

7.8 

83.9 

83.9 

1.2 

12.5 

134 

112 

1.4 

17.1 

184 

131 

1.6 

21.6 

232 

145 

1.8 

26.1 

281 

156 

2.0 

30.5 

32S 

164 

2.2 

39.4 

375 

170 

(B) 

.23 

.182 

.7 

3.3 

18.1 

25.9 

.8 

6.7 

36.8 

46.0 

.9 

10.0 

55.0 

61.1 

1.0 

13.3 

73.1 

73.1 

1.2 

20.0 

110 

91.5 

1.4 

26.6 

146 

104 

1.6 

33.2 

182 

114 

1.8 

39.9 

219 

122 

(B) 

.44 

.093 

.7 

4.  fi 

49.5 

2.0 

4.9 

g 

7.4 

79.6 

.9 

10.2 

110 

1.0 

13.2 

142 

1.2 

19.5 

210 

1.4 

26.2 

282 

1.6 

33.0 

355 

1.8 

40.0 

430 

2.0 

47.2 

508 

2.2 

54.7 

588 

2.4 

62.0 

667 

2.6 

70 

753 

ADJUSTMENT   OF   OBSERVATIONS.  77 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency — Con. 


Conditions. 

Adjusted  data. 

Probable  error 
(per  cent)  of  — 

Or. 

to 

Q 

S 

C 

ff 

E 

diusted 
data. 

Observa- 
tion. 

(B) 

0.44 

a  182 

0.5 

7.7 

42.3 

84.6 

0.9 

2.1 

.6 

11.0 

60.5 

101 

.7 

14.9 

81.9 

117 

.8 

19.0 

104 

130 

.9 

23.7 

130 

144 

1.0 

28.7 

158 

158 

1.2 

40.0 

220 

193 

1.4 

52.7 

290 

207 

1.6 

66 

363 

229 

1.8 

81 

445 

247 

(B) 

.66 

.093 

.7 

4.3 

46.2 

66.0 

2.9 

9.7 

.8 

6.2 

C6.7 

S3.  4 

.9 

8.4 

90.3 

100 

1.0 

10.8 

116 

116 

1.2 

16.3 

175 

146 

1.4 

22.9 

247 

176 

1.6 

30.0 

323 

202 

1.8 

38.0 

409 

227 

2.0 

46.5 

500 

250 

2.2 

56.0 

602 

274 

2.4 

66.0 

710 

296 

2.6 

76.5 

823 

316 

2.8 

88.0 

946 

338 

3.0 

100 

1,080 

360 

(B) 

.66 

.182 

.3 

2.8 

15.4 

51.3 

1.8 

9.7 

.4 

5.5 

30.2 

75.5 

.0 

8.9 

48.9 

97.9 

.6 

12.8 

70.4 

117 

.7 

17.3 

95.1 

136 

.8 

22.2 

122 

152 

.9 

27.6 

152 

169 

1.0 

33.5 

184 

184 

1.2 

46.7 

257 

214 

1.4 

61.4 

338 

241 

1.6 

78 

429 

268 

1.8 

96 

528 

293 

2.0 

113 

621 

310 

2.2 

134 

736 

335 

2.4 

156 

858 

357 

2.6 

179 

984 

378 

2.8 

202 

1,110 

397 

3.0 

228 

1,250 

413 

3.2 

255 

1,400 

438 

3.4 

281 

1,540 

453 

3.6 

310 

1,700 

472 

3.8 

340 

1,870- 

492 

4.0 

371 

2,040 

510 

4.2 

402 

2,210 

526 

(B) 

.66 

.363 

.5 

26.0 

71.6 

143 

1.4 

5.4 

.6 

35.2 

97 

162 

.7 

45.6 

126 

180 

.8 

56.9 

157 

198 

.9 

68.3 

188 

209 

1.0 

81 

223 

223 

1.2 

107 

295 

246 

1.4 

137 

378 

270 

1.6 

168 

463 

289 

1.8 

200 

551 

306 

2.0 

234 

644 

322 

(B) 

.66 

.545 

.2 

7.9 

14.5 

72.5 

3.6 

13.0 

.3 

17.4 

31.9 

106 

.4 

28.7 

52.6 

132 

.5 

41.3 

75.8 

152 

.6 

55.0 

101 

168 

.7 

70 

128 

183 

.8 

86 

158 

198 

.9 

102 

187 

208 

1.0 

120 

220 

220 

1.2 

158 

290 

242 

1.4 

198 

363 

259 

1.6 

240 

440 

275 

1.8 

284 

521 

289 

2.0 

330 

606 

303 

(B) 

1.00 

.182 

.5 

7.3 

40.1 

80.2 

1.1 

3.4 

.6 

10.8 

59.4 

99.0 

.7 

15.0 

82.4 

118 

.8 

19.5 

107 

134 

.9 

24.6 

135 

150 

1.0 

30.1 

165 

165 

1.2 

42.8 

235 

179 

1.4 

57 

313 

223 

1.6 

73 

401 

259 

1.8 

90 

495 

275 

2.0 

108 

594 

297 

2.2 

129 

709 

322 

2.4 

150 

824 

343 

78  TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency — Con. 


Conditions. 

Adjusted  data. 

Probable  error 
(per  cent)  of— 

Gr. 

w 

Q 

8 

C 

V 

E 

Adjusted 
data. 

Observa- 
tion. 

(B) 

1.00 

0.182 

2.6 

173 

954 

366 

1.1 

3.4 

2.8 

197 

1,080 

386 

3.0 

224 

1,230 

410 

(B) 

1.00 

.363 

.2 

3.7 

10.2 

51.0 

3.6 

16.5 

.3 

9.4 

25.9 

86.3 

.4 

16.6 

45.8 

114 

.5 

25.4 

70 

140 

.6 

35.3 

97.2 

162 

.7 

46.2 

127 

181 

.8 

58.5 

161 

201 

.9 

71 

196 

218 

1.0 

85 

234 

234 

1.2 

116 

320 

267 

1.4 

149 

411 

293 

1.6 

186 

512 

320 

1.8 

225 

620 

344 

2.0 

266 

732 

366 

2.2 

310 

854 

388 

2.4 

356 

981 

419 

(B) 

1.00 

.545 

.8 

99 

182 

228 

1.9 

6.0 

.9 

120 

220 

244 

.0 

143 

262 

262 

.2 

193 

354 

295 

.4 

247 

453 

323 

.6 

305 

560 

350 

.8 

368 

675 

375 

2.0 

435 

798 

399 

2.2 

505 

927 

421 

(B) 

1.00 

.734 

.2 

11.0 

15.0 

75 

2.8 

11.4 

.3 

25.0 

34.1 

114 

.4 

42.8 

58.3 

146 

.5 

63 

86 

172 

.6 

86 

117 

195 

.7 

110 

150 

214 

.8 

138 

188 

235 

.9 

168 

229 

254 

1.0 

199 

271 

271 

1.2 

268 

365 

304 

1.4 

344 

469 

335 

1.6 

422 

575 

359 

1.8 

507 

691 

384 

(B) 

1.32 

.182 

.4 

3.7 

20.3 

50.8 

1.6 

5.4 

.5 

6.6 

36.3 

72.6 

.6 

10.1 

55.5 

92.5 

.7 

14.1 

77.5 

111 

.8 

18.7 

103 

129 

.9 

23.8 

131 

146 

1.0 

29.3 

161 

161 

1.2 

42.0 

231 

176 

1.4 

55.8 

307 

219 

1.6 

71 

390 

244 

1.8 

88 

•  484 

269 

2.0 

105 

677 

288 

2.2 

124 

682 

310 

2.4 

144 

791 

330 

2.6 

166 

912 

351 

(B) 

1.32 

.363 

.3 

6.9 

2.4 

10.9 

.4 

13.6 

37.5 

93.8 

.5 

21.8 

60.1 

120 

.6 

31.2 

86.0 

143 

.7 

41.9 

115 

164 

.8 

53.1 

146 

182 

.9 

65.5 

180 

200 

1.0 

79 

218 

218 

1.2 

109 

300 

250 

1.4 

141 

389 

278 

1.6 

177 

488 

305 

1.8 

215 

592 

329 

2.0 

255 

702 

351 

2.2 

297 

818 

372 

(B) 

1.32 

.545 

.5 

39.8 

73 

146 

4.1 

18.3 

.6 

56 

103 

172 

.7 

75 

138 

197 

.8 

95 

174 

218 

.9 

116 

213 

237 

1.0 

140 

257 

257 

1.2 

191 

351 

292 

1.4 

248 

455 

325 

1.6 

310 

5«9 

355 

1.8 

375 

688 

382 

(B) 

1.32 

.734 

.3 

21.7 

29.6 

98.7 

2.1 

9.7 

.4 

39.0 

53.1 

133 

.5 

59.5 

81.0 

162 

.6 

83 

113 

188 

.7 

110 

150 

2U 

ADJUSTMENT   OF   OBSERVATIONS.  79 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency — Con. 


.     Conditions. 

Adjusted  data. 

Probable  error 
(per  cent)  of— 

Or. 

w 

Q 

S 

C 

U 

K 

\tljusted 
data. 

Observa- 
tion. 

(B) 

1.32 

0.734 

0.8 

139 

189 

236 

2.1 

9.7 

.9 

170 

232 

258 

1.0 

204 

278 

278 

1.2 

279 

380 

317 

1.4 

361   • 

492 

351 

1.6 

450 

613 

383 

1.8 

545 

742 

412 

(B) 

1.96 

.363 

.4 

7.7 

21.2 

50.3 

3.9 

15.5 

.5 

14.3 

39.4 

76.8 

.6 

22.4 

61.7 

103 

.7 

32.0 

88.2 

126 

.8 

43.0 

118 

148 

.9 

54.8 

151 

168 

1.0 

67.6 

186 

186 

1.2 

97 

267 

222 

1.4 

130 

358 

256 

1.6 

167 

460 

287 

1.8 

208 

573 

318 

2.0 

253 

697 

348 

2.2 

300 

826 

375 

(B) 

1.96 

.545 

.5 

29.8 

54.7 

109 

2.9 

11.3 

.6 

44.2 

81.1 

135 

.7 

60.5 

111 

159 

.8 

79 

145 

181 

.9 

98 

180 

200 

1.0 

120 

220 

220 

1.2 

168 

308 

257 

1.4 

221 

406 

290 

1.6 

282 

517 

323 

1.8 

345 

633 

352 

(B) 

1.96 

.734 

.3 

18.6 

25.3 

84.3 

4.1 

17.1 

.4 

35.8 

48.8 

122 

.5 

56.0 

76.3 

155 

.6 

78.5 

107 

178 

.7 

103 

140 

200 

.8 

130 

177 

221 

.9 

158 

215 

239 

1.0 

190 

259 

259 

1.2 

257 

350 

292 

1.4 

330 

450 

322 

1.6 

407 

554 

346 

1.8 

488 

665 

369 

2.0 

575 

783 

392 

(B) 

1.96 

1.119 

.2 

13.4 

12.0 

60 

5.4 

24.1 

.3 

36.1 

32.3 

108 

.4 

65.0 

58.1 

145 

.5 

98 

87.5 

175 

.6 

135 

121 

202 

.  7 

175 

156 

223 

.8 

218 

195 

241 

.9 

263 

235 

261 

1.0 

313 

280 

280 

1.2 

417 

373 

311 

(C) 

.44 

.093 

.6 

2.3 

24.7 

41.2 

3.3 

10.1 

.  7 

4.1 

44.1 

63.0 

.8 

6.1 

65.6 

82.0 

.9 

8.4 

90.4 

100 

1.0 

10.9 

117 

117 

1.2 

16.4 

176 

147 

1.4 

22.6 

243 

174 

1.6 

29.4 

316 

197 

1.8 

36.6 

394 

219 

2.0 

44.2 

475 

238 

2.2 

52.3 

562 

255 

(C) 

.44 

.182 

.6 

8.7 

47.8 

79.7 

2.3 

6.1 

.7 

12.1 

66.5 

95 

.8 

15.9 

87.4 

109 

.9 

20.1 

110 

122 

1.0 

24.7 

136 

136 

1.2 

34.3 

188 

157 

1.4 

45.3 

249 

178 

1.6 

57.2 

314 

196 

1.8 

70 

385 

214 

2.0 

84 

462 

231 

2.2 

98 

538 

244 

(C) 

.66 

.093 

.5 

3.3 

35.5 

71.0 

1.7 

6.8 

.6 

5.0 

53.8 

89.7 

.7 

6.9 

74.2 

106 

.8 

9.0 

96.8 

121 

.9 

11.3 

121 

134 

1.0 

13.8 

148 

148 

1.2 

19.4 

209 

174 

1.4 

25.7 

276 

197 

1.6 

32.4 

348 

217 

1.8 

39.7 

427 

237 

80  TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  viith  corresponding  values  of  duty  and  efficiency — Con. 


Conditions. 

Adjusted  data. 

Probable  error 
(per  cent)  of— 

Or. 

w 

Q 

S 

C 

U 

E 

Adjusted 
data. 

Observa- 
tion. 

(C) 

0.66 

0.093 

2.0 

47.5 

511 

256 

1.7 

6.8 

2.2 

56 

602 

274 

2.4 

65 

699 

291 

2.6 

74 

796 

306 

2.8 

84 

904 

323 

3 

95 

1,020 

340 

3.2 

105 

,130 

353 

3.4 

117 

,2fiO 

371 

3.6 

128 

,380 

384 

3.8 

140 

,510 

398 

4 

153 

,650 

412 

4.2 

166 

,780 

424 

(C) 

.66 

.182 

.3 

3 

16.5 

55 

1.2 

5.8 

.4 

5.7 

31.3 

75.8 

.5 

9 

49.5 

99 

.6 

12.9 

70.9 

118 

.7 

17.1 

94 

134 

.8 

21.8 

120 

150 

.9 

26.7 

147 

163 

1 

32.1 

176 

176 

1.2 

44.1 

242 

202 

1.4 

57 

313 

223 

1.6 

71 

390 

244 

1.8 

87 

478 

265 

2 

102 

5M) 

280 

2.2 

120 

659 

299 

2.4 

138 

758 

316 

2.6 

157 

863 

332 

2.8 

177 

892 

347 

3 

198 

,090 

363 

3.2 

219 

,200 

375 

3.4 

241 

,320 

388 

3.6 

2fi6 

,460 

406 

3.8 

289 

,590 

419 

(C) 

.66 

.363 

.4 

16.5 

45.5 

114 

1.8 

9.0 

.5 

23.3 

64.2 

128 

.6 

31.9 

87.9 

146 

.7 

41.2 

113 

161 

.8 

51.1 

141 

176 

.9 

62 

171 

190 

1 

73 

201 

201 

1.2 

98 

270 

225 

1.4 

124 

342 

244 

1.6 

152 

419 

262 

1.8 

183 

514 

280 

2 

215 

59? 

296 

2.2 

MS 

683 

310 

2.4 

2S3 

780 

325 

2.6 

320 

882 

339 

(C) 

.66 

.545 

.2 

6.6 

12.1 

60.5 

2.1 

10.1 

.3 

14.8 

27.2 

90.7 

.4 

24.8 

45.5 

114 

.5 

36.5 

67.0 

134 

.6 

49.2 

90.2 

150 

.7 

63.3 

116 

166 

.8 

78.5 

144 

180 

.9 

95 

174 

193 

1 

112 

205 

205 

1.2 

150 

275 

229 

1.4 

190 

349 

249 

1.6 

233 

427 

267 

1.8 

280 

614 

285 

2 

328 

M>2 

301 

2.2 

380 

697 

317 

(C) 

.66 

.734 

.5 

51 

69.5 

139 

3.8 

8.5 

.6 

68 

93 

154 

.7 

87 

118 

169 

.8 

107 

146 

182 

.9 

129 

176 

196 

1 

152 

207 

207 

(C) 

1.00 

.182 

.4 

4.5 

24.7 

61.8 

1 

4.4 

.5 

7.6 

41.8 

83.6 

.6 

11.4 

62.6 

104 

.7 

15.6 

85.8 

123 

.8 

20.4 

112 

140 

.9 

25.6 

141 

157 

1 

31.2 

171 

171 

1.2 

44 

242 

202 

1.4 

57.7 

317 

226 

1.6 

72 

396 

247 

1.8 

90 

495 

275 

2 

106 

582 

291 

2.2 

125 

686 

312 

2.4 

145 

797 

332 

2.6 

166 

912 

351 

2.8 

189 

1,040 

372 

ADJUSTMENT   OF   OBSERVATIONS.  81 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency — Con. 


Conditions. 

Adjusted  data. 

Probable  error 
(per  cent)  of— 

Or. 

w 

Q 

S 

C 

V 

E 

Adjusted 
data. 

Observa- 
tion. 

(C) 

1.00 

0.363 

0.2 

3.3 

9.1 

45.5 

1.2 

6.2 

.3 

9.6 

26.4 

88 

.4 

17.5 

48.2 

120 

.5 

26.6 

73.3 

147 

.6 

36.6 

101 

168 

.7 

47.5 

131 

187 

.8 

59 

1(8 

204 

.9 

72 

198 

220 

1 

85 

234 

234 

1.2 

113 

311 

259 

1.4 

143 

394 

281 

1.6 

175 

482 

301 

1.8 

210 

579 

322 

2 

245 

675 

338 

2.2 

281 

774 

352 

2.4 

320 

882 

367 

2.6 

3fiO 

992 

381 

2.8 

401 

1,100 

393 

(C) 

1.00 

.545 

.5 

48.2 

88.4 

177 

1.1 

5.5 

.6 

64.9 

119 

19S 

.7 

82 

150 

214 

.8 

100 

183 

229 

.9 

119 

218 

242 

1.0 

140 

257 

257 

1.2 

183 

336 

280 

1.4 

228 

418 

299 

1.6 

275 

504 

315 

1.8 

326 

598 

332 

2.0 

377 

692 

346 

2.2 

430 

789 

359 

(C) 

1.00 

.734 

.2 

12.5 

17.0 

85 

7.8 

37.7 

.3 

27.3 

37.2 

124 

.4 

44.2 

60.2 

150 

.5 

63.2 

86.2 

172 

.6 

84 

114 

190 

.7 

106 

144 

206 

.8 

130 

177 

221 

.9 

155 

211 

234 

1.0 

ISO 

245 

245 

1.2 

235 

320 

267 

1.4 

294 

401 

286 

1.6 

352 

480 

300 

1.8 

416 

567 

315 

2.0 

483 

658 

329 

2.2 

550 

750 

341 

(C) 

1.00 

1.119 

.5 

96 

85.8 

172 

2.3 

5.1 

.6 

128 

114 

190 

.7 

162 

145 

207 

.8 

198 

177 

221 

.9 

236 

211 

234 

1.0 

276 

247 

247 

1.2 

361 

323 

269 

1  4 

451 

403 

288 

(C) 

1.32 

.182 

.6 

5.6 

30.8 

51.3 

2.4 

S.  4 

.7 

8.7 

47.8 

68.3 

.8 

12.3 

67.6 

84.5 

.9 

16.6 

91.2 

101 

1.0 

21.4 

118 

118 

1.2 

32.7 

180 

150 

1.4 

46.5 

256 

183 

1.6 

62 

341 

213 

1.8 

80 

440 

244 

2.0 

99 

544 

272 

2.2 

120 

659 

299 

2.4 

144 

791 

329 

(C) 

1.32 

.363 

.3 

6.4 

17.6 

58.7 

1.2 

5.5 

.4 

13.6 

37.5 

93.8 

.5 

22.1 

60.9 

122 

.6 

31.9 

87.9 

146 

.7 

42.2 

116 

166 

.8 

53.8 

148 

185 

.9 

66 

182 

202 

1.0 

79 

218 

218 

1.2 

105 

289 

241 

1.4 

135 

372 

266 

1.6 

166 

458 

286 

1.8 

199 

548 

304 

2.0 

233 

642 

321 

2.2 

270 

744 

338 

2.4 

307 

846 

352 

(C) 

1.32 

.545 

.6 

56.3 

103 

172 

1.8 

8.5 

.  7 

73.3 

134 

191 

.8 

91 

167 

209 

.9 

109 

200 

222 

1.0 

129 

237 

237 

1.2 

171 

314 

262 

20921°— No.  86— 14 


82  TRANSPORTATION   OF   DEBRIS   BY   RUNNING    WATER. 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency — Con. 


Conditions. 

Adjusted  data. 

Probable  error 
(per  cent)  of— 

Or. 

w 

Q 

S 

C 

V 

E 

\djusted 
data. 

Observa- 
tion. 

(C) 

1.32 

0.345 

1.4 

215 

395 

282 

1.8 

8.3 

1.6 

263 

483 

302 

1.8 

312 

573 

318 

2.0 

362 

664 

332 

2.2 

415 

762 

346 

2.4 

470 

862 

359 

(C) 

1.32 

.734 

.2 

9.3 

12.7 

63.5 

2.2 

9.0 

.3 

25.0 

34.1 

114 

.4 

43.2 

58.9 

147 

.5 

63.8 

86.9 

174 

.6 

86 

117 

195 

.7 

109 

148 

211 

.8 

133 

181 

226 

.9 

160 

218 

242 

1.0 

187 

255 

255 

1.2 

243 

331 

276 

1.4 

304 

414 

296 

1.6 

367 

500 

312 

1.8 

432 

589 

327 

2.0 

500 

681 

340 

2.2 

870 

777 

353 

(C) 

1.96 

.363 

.5 

10.5 

28.9 

57.8 

3.5 

15.5 

.6 

17.8 

49.0 

81.7 

.7 

26.5 

73.0 

104 

.8 

36.6 

101 

126 

.9 

47.9 

132 

147 

1.0 

60.0 

165 

165 

1.2 

88 

242 

202 

1.4 

119 

328 

234 

1.6 

154 

424 

265 

1.8 

193 

532 

295 

2.0 

235 

648 

324 

2.2 

280 

771 

350 

(C) 

1.96 

.545 

.5 

25.7 

47.2 

94.4 

2.0 

7.0 

.6 

39.3 

72.1 

120 

.7 

55.0 

101 

144 

.8 

72 

132 

165 

.9 

91 

167 

186 

1.0 

111 

204 

204 

1.2 

156 

286 

238 

1.4 

205 

376 

268 

1.6 

259 

475 

297 

1.8 

316 

580 

322 

2.0 

375 

688 

344 

2.2 

440 

808 

367 

(C) 

1.96 

.734 

.4 

34.0 

46.3 

116 

2.8 

13.2 

.5 

55 

74.9 

150 

.6 

78 

106 

177 

.7 

103 

140 

200 

.8 

131 

178 

222 

.9 

160 

218 

242 

1.0 

190 

259 

259 

1.2 

255 

347 

289 

1.4 

323 

410 

314 

1.6 

394 

537 

336 

(("> 

1.% 

1.119 

.5 

88 

78.7 

157 

2.2 

P.  9 

.fi 

130 

116 

193 

.7 

177 

158 

226 

.8 

228 

204 

255 

.9 

283 

253 

281 

1.0 

343 

307 

307 

1.2 

480 

429 

358 

(D) 

.66 

.093 

.6 

2.7 

29.0 

48.4 

2.6 

8.1 

.7 

3.9 

41.9 

59.9 

.8 

5.4 

58.1 

72.5 

.9 

7.1 

70.4 

84.9 

1.0 

9.1 

97.9 

97.9 

1.2 

13.5 

145 

121 

1.4 

18.6 

200 

143 

1.6 

24.7 

266 

166 

1.8 

31.4 

338 

188 

2.0 

38.5 

414 

207 

2.2 

46.3 

498 

226 

2.4 

55.2 

594 

248 

2.6 

64.3 

692 

266 

(D) 

.66 

.182 

.4 

4.7 

25.8 

64.5 

4.2 

15.2 

.5 

7.7 

42.3 

84.6 

.6 

11.3 

62.1 

104 

.7 

15.3 

84.1 

120 

.8 

19.7 

108 

135 

.9 

24.6 

135 

150 

1.0 

29.8 

•  164 

164 

1.2 

41.0 

235 

188 

1.4 

53.4 

293 

209 

ADJUSTMENT   OF   OBSERVATIONS.  83 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency— Con. 


Conditions. 

Adjusted  data. 

Probable  error 
(per  cent)  of— 

Or. 

w 

Q 

S 

C 

U 

E 

Adjusted 
data. 

observa- 
tion. 

(D) 

0.66 

0.182 

1.6 

67 

368 

230 

4.2 

15.2 

1.8 

82 

450 

250 

2.0 

98 

538 

269 

2.2 

114 

626 

284 

2.4 

132 

726 

302 

2.6 

150 

824 

317 

(D) 

.6f> 

.545 

.2 

4.7 

8.5 

42.5 

3.4 

15.3 

.3 

11.7 

21.5 

71.6 

.4 

20.7 

38.0 

95 

.5 

31.2 

57.2 

114 

.6 

43.1 

79.1 

132 

.7 

56.2 

103 

147 

.8 

70.2 

129 

161 

.9 

86 

158 

176 

1.0 

101 

1S5 

185 

1.2 

137 

251 

209 

1.4 

175 

321 

229 

1.6 

217 

398 

249 

1.8 

261 

479 

266 

2.0 

308 

568 

284 

2.2 

357 

655 

298 

(D) 

1.00 

.182 

.7 

11.3 

62.1 

88.7 

2.5 

9.1 

.8 

15.1 

83 

104 

.9 

19.2 

106 

118 

1.0 

24.0 

132 

132 

1.2 

34.5 

190 

158 

1.4 

46.4 

255 

182 

1.6 

59.8 

329 

206 

1.8 

74 

407 

226 

2.0 

90 

495 

248 

2.2 

106 

582 

264 

2.4 

124 

682 

284 

2.6 

144 

792 

304 

2.8 

164 

902 

324 

(D) 

1.00 

.363 

.3 

6.9 

19.0 

63 

3.3 

13.2 

.4 

13.2 

36.4 

91.0 

.5 

20.9 

57.6 

115 

.6 

29.6 

81.6 

136 

.7 

39.0 

107 

153 

.8 

49.6 

137 

171 

.9 

61 

168 

187 

1.0 

73 

201 

201 

1.2 

99 

273 

228 

1.4 

127 

350 

250 

1.6 

158 

435 

272 

1.8 

190 

524 

291 

2.0 

225 

620 

310 

2.2 

261 

819 

327 

2.4 

299 

824 

343 

(D) 

1.00 

.545 

.5 

27.6 

50.6 

101 

1.2 

3.9 

.6 

40.2 

73.8 

123 

.7 

55.0 

101 

144 

.8 

71 

130 

162 

.9 

89 

163 

181 

1.0 

108 

'  198 

198 

1.2 

152 

279 

232 

1.4 

201 

369 

264 

1.6 

257 

472 

295 

1.8 

317 

582 

323 

2.0 

382 

700 

350 

(D) 

1.00 

.734 

.2 

6.9 

9.4 

47.0 

2.5 

11.5 

.3 

17.5 

23.8 

79.3 

.4 

31.0 

42.2 

106 

.5 

47.0 

64.0 

128 

.6 

65.0 

88.6 

148 

.7 

85 

116 

166 

.8 

106 

144 

180 

.9 

129 

176 

196 

1.0 

153 

208 

208 

1.2 

208 

282 

236 

1.4 

268 

365 

361 

1.6 

332 

452 

283 

1.8 

388 

542 

302 

(D) 

1.32 

.363 

.5 

11.9 

32.8 

65.6 

7.0 

24.3 

.6 

18.8 

50.9 

84.8 

.7 

27.0 

74.4 

106 

.8 

36.6 

101 

139 

.9 

47.2 

130 

144 

1.0 

59.2 

163 

163 

1.2 

87 

240 

200 

1.4 

118 

325 

232 

1.6 

154 

424 

265 

1.8 

194 

534 

297 

2.0 

237 

653 

326 

84  TRANSPORTATION    OF    DEBRIS    BY    RUNNING    WATER. 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency — Con. 


Conditions. 

Adjusted  data. 

Probable  error 
(percent)  of  — 

Gr. 

VI 

Q 

S 

C 

U 

E 

Adjusted 
data. 

Observa- 
tion. 

(D) 

1.32 

0.734 

0.3 

10.2 

13.9 

46.3 

5.3 

21.4 

.4 

20.9 

28.5 

71.2 

.5 

34.0 

46.3 

92.6 

.6 

49.2 

67.0 

112 

.  7 

67 

91.3 

130 

.8 

87 

118 

148 

.9 

108 

147 

163 

1.0 

131 

178 

178 

1.2 

185 

252 

210 

(E) 

.06 

.363 

1.0 

24.8 

68.3 

68.3 

0.9 

2.1 

1.2 

36.2 

99.7 

83.1 

1.4 

49.4 

136 

97.2 

(E) 

.66 

.734 

1.0 

40.0 

54.5 

54.5 

1.2 

59.0 

80.4 

67.0 

1.4 

82 

112 

80 

1.6 

108 

147 

92 

(E) 

1.00 

.182 

.4 

2.4 

13.2 

33.0 

1.9 

5.0 

.5 

3.8 

20.9 

41.8 

.6 

5.5 

30.4 

50.7 

.7 

7.4 

40.6 

58.0 

.8 

9.6 

52.7 

65.0 

.9 

12.0 

66.0 

73.3 

1.0 

14.8 

81.4 

81.4 

1.2 

20.7 

114 

95 

1.4 

27.6 

152 

108 

1.6 

35.3 

194 

121 

1.8 

43.5 

239 

133 

2.0 

52.5 

288 

144 

2.2 

62.4 

343 

156 

2.4 

73 

401 

167 

2.6 

84 

461 

177 

2.8 

96 

527 

188 

(E) 

1.00 

.363 

.2 

1.3 

3.6 

18.0 

5.1 

14.6 

.3 

3.3 

9.1 

30.3 

.4 

6.0 

16.5 

41.2 

.5 

9.2 

25.3 

50.6 

.6 

13.1 

36.1 

60.1 

.7 

17.5 

48.2 

68.9 

.8 

22.5 

62.0 

77.5 

.9 

28.0 

77.2 

85.8 

1.0 

33.8 

93.1 

93.1 

1.2 

47.2 

130 

108 

1.4 

62.0 

171 

122 

1.6 

78.5 

216 

135 

1.8 

96 

264 

147 

2.0 

115 

317 

158 

2.2 

137 

378 

172 

2.4 

159 

438 

182 

(E) 

1.00 

.734 

.2 

4.2 

5.7 

28.5 

5.0 

19.5 

.3 

9.1 

12.4 

41.3 

.4 

15.2 

20.7 

51.8 

.5 

22.4 

30.5 

61.0 

.6 

30.8 

42.0 

70.0 

.7 

40.0 

54.5 

77.9 

.8 

49.8 

67.8 

84.8 

.9 

60.5 

72.4 

91.6 

1.0 

72.0 

98.1 

98.1 

1.2 

98 

134 

112 

1.4 

126 

172 

123 

1.6 

156 

213 

133 

1.8 

188 

256 

142 

2.0 

222 

302 

III 

(E) 

1.00 

1.119 

.5 

44 

39.3 

76.6 

3.2 

6.4 

.6 

60 

54 

89 

.7 

77 

69 

98 

.8 

97 

87 

108 

.9 

117 

105 

117 

1.0 

138 

123 

123 

1.2 

188 

168 

140 

1.4 

242 

216 

154 

(E) 

1.32 

.363 

.6 

15.3 

42.2 

70.3 

.7 

20.3 

55.9 

79.9 

.8 

25.4 

70.0 

87.5 

.9 

30.7 

84.6 

94.0 

1.0 

36.3 

100 

100 

1.2 

49.3 

136 

113 

1.4 

63.5 

175 

125 

1.6 

78.8 

217 

136 

1.8 

95 

262 

145 

2.0 

112 

309 

154 

(E) 

1.32 

.734 

.5 

24.7 

33.7 

67.4 

.6 

33.0 

45.0 

75.0 

.7 

42.0 

57.2 

81.7 

.8 

51.7 

70.4 

8S.O 

.9 

62 

84.5 

93.9 

1.0 

73 

99.5 

99.5 

1.2 

97 

132 

110 

ADJUSTMENT   OF   OBSERVATIONS.  85 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency — Con. 


Conditions. 

Adjusted  data. 

Probable  error 
(per  cent)  o(— 

Or. 

w 

Q 

8 

C 

U 

M 

Adjusted 
data. 

Observa- 
tion. 

(E) 

1.32 

1.119 

0.6 

49.4 

44.2 

73.7 

1.8 

3.6 

.7 

65.0 

58.1 

83.0 

.8 

82.5 

73.7 

92.1 

.9 

102 

91.1 

101 

1.0 

123 

110 

110 

1.2 

171 

153 

128 

(F) 

.66 

.182 

1.2 

9.3 

51.0 

42.5 

.8 

1.7 

1.4 

13.7 

75.3 

53.8 

1.6 

18.9 

104 

65.0 

1.8 

24.7 

136 

75.6 

2.0 

31.4 

172 

86.0 

2.2 

38.7 

213 

96.8 

2.4 

46.8 

257 

107 

2.6 

55.7 

306 

118 

(F) 

.66 

.363 

1.0 

20.5 

56.5 

56.5 

2.3 

4.6 

1.2 

29.9 

82.4 

68.7 

1.4 

41.0 

113 

80.8 

1.6 

53.2 

147 

91.9 

1.8 

67.0 

185 

103 

2.0 

81.5 

225 

112 

(F) 

.66 

.734 

.8 

31.6 

43.0 

47.8 

.4 

.8 

1.0 

39.0 

53.2 

53.2 

1.2 

55.3 

75.3 

62.8 

1.4 

73.0 

99.5 

71.0 

1.6 

93.5 

127 

79.4 

1.8 

115 

157 

87.2 

(F) 

1.00 

.182 

1.2 

4.2 

23.1 

17.6 

1.4 

7.3 

40.1 

28.6 

1.6 

11.3 

62.1 

38.8 

1.8 

16.4 

90.1 

50.0 

2.0 

22.5 

124 

62.0 

2.2 

29.8 

164 

74.6 

2.4 

38.2 

210 

87.5 

2.6 

47.7 

262 

101 

(F) 

1.00 

.363 

.8 

7.4 

20.4 

25.5 

2.0 

4.8 

.9 

10.5 

28.9 

32.1 

1.0 

14.0 

38.6 

38.6 

1.2 

22.8 

62.8 

52.3 

1.4 

33.4 

92.0 

65.7 

1.6 

46.0 

127 

79.4 

1.8 

60 

165 

91.7 

2.0 

76 

209 

104 

2.2 

94 

259 

118 

2.4 

112 

309 

129 

2.6 

133 

367 

141 

(F) 

1.00 

.734 

.7 

20.0 

27.3 

39.0 

.5 

1.0 

.8 

27.3 

37.2 

46.5 

.9 

35.1 

47.8 

53.1 

1.0 

42.8 

58.3 

58.3 

1.2 

63 

85.8 

71.5 

1.4 

85 

116 

82.9 

1.6 

109 

148 

92.6 

1.8 

136 

185 

103 

(F) 

1.00 

1.119 

.7 

37.5 

33.5 

47.9 

.9 

1.8 

1 

.8 

51.0 

45.6 

57.0 

.9 

64.8 

57.9 

64.3 

1.0 

79.5 

71.0 

71.0 

1.2 

112 

100 

83.3 

1.4 

150 

134 

95.8 

1.6 

191 

171 

107 

1.8 

236 

211 

117 

(F) 

1.32 

.363 

1.2 

21.4 

59.0 

48.2 

l72 

2.4 

1.4 

31.0 

85.4 

61.0 

1.6 

41.5 

114 

71.3 

1.8 

53.0 

146 

81.1 

2.0 

66.0 

182 

91.0 

2.2 

80 

220 

100 

(F) 

1.32 

.734 

.8 

22.9 

29.6 

?7.0 

3.2 

7.2 

.9 

31.0 

40.1 

44.6 

1.0 

40.2 

52.0 

52.0 

1.2 

61.6 

79.7 

66.4 

1.4 

87.0 

119 

85.0 

1.6 

115 

149 

93.2 

(F) 

1.32 

1.119 

.8 

43.3 

38.7 

48.4 

3.2 

7.2 

.9 

56.0 

50.0 

91.1 

1.0 

70.0 

62.6 

62.6 

1.2 

101 

90.2 

75.2 

1.4 

138 

123 

87.9 

1.6 

178 

159 

99.4 

1.8 

221 

197 

109 

86  TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency — Con. 


Conditions. 

Adjusted  data. 

Probable  error 
(per  cent)  of— 

Gr. 

w 

Q 

S 

C 

V 

X 

Adjusted 
data. 

Observa- 
tion. 

(G) 

0.66 

0.363 

.0 

8.5 

23.4 

23.4 

2,2 

5.8 

.2 

14.8 

40.8 

34.0 

.4 

23.0 

63.4 

45.3 

.6 

32.7 

90.1 

56.3 

.8 

43.7 

120 

66.7 

2.0 

56.0 

154 

77.0 

(G) 

.66 

.734 

.  7 

10.3 

13.3 

19.0 

1.5 

4.3 

.8 

16.0 

20.7 

25.9 

.9 

22.8 

29.5 

32.8 

.0 

30.4 

39.3 

39.3 

.2 

48.3 

62.5 

52.1 

.4 

69 

89.2 

63.7 

.6 

93 

127 

79.4 

.8 

120 

155 

86.2 

2.0 

149 

193 

96.5 

2.2 

181 

234 

106 

2.4 

216 

280 

117 

2.6 

253 

327 

126 

(G) 

.66 

1.119 

.6 

11.9 

10.6 

17.7 

1.2 

3.6 

.7 

19.0 

17.0 

24.3 

.8 

27.7 

24.7 

30.9 

.9 

37.6 

33.6 

37.3 

1.0 

49.0 

43.8 

43.8 

1.2 

75.0 

67.0 

55.8 

1.4 

107 

95.6 

68.3 

1.6 

143 

128 

80.0 

1.8 

182 

163 

90.6 

2.0 

227 

203 

102 

2.2 

276 

247 

112 

2.4 

325 

290 

121 

2.6 

382 

341 

131 

(G) 

1.00 

.363 

1.2 

9.7 

26.7 

22.2 

1.9 

5.0 

1.4 

16.3 

44.9 

35.0 

1.6 

24.4 

67.2 

42.0 

1.8 

34.1 

94.0 

52.2 

2.0 

45.2 

125 

62.5 

2.2 

57 

157 

71.4 

2.4 

71 

196 

81.7 

2.6 

86 

237 

91.2 

2.8 

102 

281 

100 

3.0 

120 

331 

no 

(G) 

1.00 

.734 

.7 

8.0 

10.9 

15.6 

1.9 

5.5 

.8 

13.3 

18.1 

22.6 

.9 

19.5 

26.6 

29.6 

1.0 

26.9 

36.6 

36.6 

1.2 

44 

60.0 

50.0 

1.4 

65 

88.6 

63.3 

1.6 

89 

121 

75.6 

1.8 

115 

157 

87.8 

2.0 

145 

198 

99.0 

2.2 

177 

241 

110 

2.4 

212 

289 

120 

2.6 

249 

339 

130 

(G) 

1.00 

1.119 

.6 

13.5 

12.1 

20.1 

4.3 

13.6 

.7 

22.7 

20.3 

29.0 

.8 

33.6 

30.0 

37.5 

.9 

46.0 

41.1 

45.6 

1.0 

61.0 

54.4 

54.4 

1.2 

92 

82.2 

68.5 

1.4 

128 

114 

81.5 

1.6 

171 

153 

95.6 

1.8 

217 

194 

108 

2.0 

268 

239 

120 

2.2 

321 

287 

130 

2.4 

380 

340 

142 

(G) 

1.32 

.363 

1.8 

18.1 

49.5 

27.7 

2.4 

4.8 

2.0 

28.3 

78.0 

39.0 

2.2 

39.9 

101 

45.9 

2.4 

53.8 

148 

61.7 

2.6 

69.8 

192 

73.9 

2.8 

88.0 

242 

86.5 

3.0 

109 

300 

100 

3.2 

134 

389 

115 

(G) 

1.32 

.734 

.8 

7.9 

10.8 

13.5 

2.2 

5.8 

.9 

12.8 

17.4 

19.3 

1.0 

18.9 

25.8 

25.8 

1.2 

33.8 

46.1 

38.4 

1.4 

52.0 

70.8 

50.6 

1.6 

74.0 

101 

63.2 

1.8 

98.0 

134 

75.4 

2.0 

125 

170 

85.0 

2.2 

155 

211 

95.9 

2.4 

187 

255 

106 

2.6 

222 

303 

117 

2.8 

260 

354 

126 

ADJUSTMENT   OF   OBSERVATIONS.  87 

TABLE  12. — Adjusted  values  of  capacity,  based  on  data  of  Table  4,  with  corresponding  values  of  duty  and  efficiency— Con. 


Conditions. 

Adjusted  data. 

Probable  error 
(per  cent)  of— 

Or. 

• 

Q 

S 

C 

U 

E 

Adjusted 
data. 

Observa- 
tion. 

(0) 

1.32 

1.119 

0.7 

17.9 

16.0 

22.9 

3.1 

8.7 

.8 

28.0 

25.0 

31.2 

.9 

39.5 

35.3 

38.1 

1.0 

52.2 

46.6 

40.  6 

1.2 

81 

72.4 

60.3 

1.4 

115 

103 

73.6 

1.6 

153 

137 

85.6 

1.8 

194 

173 

96.2 

2.0 

238 

213 

108 

2.2 

285 

255 

116 

2.4 

335 

299 

125 

2.6 

387 

346 

133 

(H) 

.66 

.363 

1.2 

3.1 

8.5 

7.1 

1.0 

3.5 

1.4 

6.9 

19.0 

13.6 

1.6 

12.4 

34.2 

21.4 

1.8 

19.4 

53.5 

29.7 

2.0 

28.0 

77.1 

38.6 

2.2 

38.1 

105 

47.8 

2.4 

49.8 

137 

57.1 

2.6 

63 

174 

66.9 

2.8 

78 

215 

76.8 

(H) 

.66 

.734 

.8 

4.5 

6.8 

8.5 

1.6 

4.9 

.9 

8.2 

12.4 

13.8 

.0 

12.8 

19.4 

19.4 

.2 

24.6 

37.3 

31.1 

.4 

39.2 

59.4 

42.4 

.6 

57.0 

86.4 

54.0 

.8 

77.0 

117 

65.0 

2.0 

100 

152 

76.0 

2.2 

125 

189 

85.9 

2.4 

151 

229 

95.6 

2.6 

181 

274 

105 

2.8 

213 

323 

115 

3.0 

246 

373 

126 

3.2 

282 

427 

133 

(H) 

.66 

1.119 

.7 

6.0 

5.4 

7.7 

11.2 

.8 

11.1 

9.9 

12.4 

3.2 

.9 

17.6 

15.7 

17.4 

1.0 

25.2 

22.5 

22.5 

1.2 

43.5 

38.8 

32.3 

1.4 

65.3 

58.3 

41.6 

1.6 

91.0 

81.3 

50.8 

1.8 

120 

107 

59.5 

2.0 

152 

136 

68.0 

2.2 

186 

166 

75.5 

2.4 

223 

199 

83.0 

2.6 

263 

235 

90.4 

OBSERVATIONS  ON  DEPTH. 

MODE   OF    ADJUSTMENT. 

To  adjust  the  observations  of  depth  it  is 
necessary  to  deal  with  their  relations  to  another 
variable,  and  either  slope  or  capacity  might  be 
used.  The  selection  of  the  particular  variable 
for  comparison  was  a  matter  of  convenience 
only,  because  adjustment  to  either  one  would 
bring  the  depth  values  into  orderly  relation  to 
the  adjusted  values  of  the  other  also;  but  the 
question  of  convenience  was  not  unimportant. 
A  fairly  thorough  preliminary  study  was  there- 
fore made,  in  which  the  depth  measurements 
for  many  series  were  plotted  in  relation,  sever- 
ally, to  measurements  of  capacity  and  measure- 
ments of  slope. 

In  figure  27  the  horizontal  scale  is  that  of 
slope,  the  vertical  of  depth.  The  round  dots 
represent  observations  made  with  grade  (C), 


width  1.32  feet,  and  discharge  0.734  ft.3/sec. 
Despite  irregularities,  the  grouping  suggests  as 


+ 

1^ 

\^^  * 

IS.      + 

\ 
\ 

s*. 

\ 

\ 

*"u-«  - 

'-•—  • 

0.5                   1.0                   1.5 
Slope 

FIGUKE  27.— Observations  of  depth  of  current,  in  relation  to  slope, 
plotted  as  dots.  The  crosses  show  logarithms  of  the  same  depths 
and  slopes. 

the  representative  lino  some  such  curve  as  that 
drawn.     On  taking  account  of  the  physical  con- 


88 


TRANSPORTATION    OF    DEBRIS   BY   RUNNING    WATER. 


ditions,  it  is  evident  that  as  the  slope  is  flat- 
tened, the  current  is  slowed  and  the  depth  in- 
creased, and  that  zero  slope  gives  infinite  depth. 
The  theoretic  curve,  therefore,  has  an  asymp- 
tote in  the  vertical  line  corresponding  to  zero 
slope.  Similarly  the  depth  is  reduced  by  in- 
crease in  slope  but  remains  finite  for  very  high 
slopes.  The  theoretic  curve  has  as  asymptote 
a  horizontal  line  corresponding,  exactly  or 
appioximately,  with  the  (horizontal)  line  of 
zero  depth.  These  asymptotes  relate  the  curve 
d=f(S)  to  the  hyperbola.  In  the  same  figure, 
but  with  use  of  a  different  scale,  are  a  series  of 
crosses  which  show  the  same  observations  as 
they  appear  when  plotted  on  logarithmic  sec- 
tion paper.  They  are  the  plot  of  observations 
ou  log  d  =/,  (log  8) ;  and  their  arrangement  sug- 
gests that  the  representative  line  may  be 
straight. 

Plots  were  made  to  show  the  relations  of 
depth  observations  to  associated  capacity,  and 
these  also  suggest  the  hyperbola  and  the 
straight  line.  If,  however,  the  locus  of  d  =/u(  f) 
is  a  hyperbola  it  differs  materially  from  that  of 
d=f(S),  for  as  depth  increases  and  current 
slackens,  capacity  becomes  zero  when  current 
reaches  the  value  of  competence,  and  depth 
is  not  then  infinite.  So  the  line  of  zero  capac- 
ity is  not  an  asymptote  to  the  curve. 

It  is  to  be  observed  also  that  the  represen- 
tative lines  for  log  d  =/,  (log  S)  and  log  d  =/„ 
Gog  (7)  can  not  both  be  straight,  for  if  they  were 
there  could  be  derived  from  them  a  straight 
line  representing  log  C=fv  (log  S),  and  it  has 
already  been  found  that  that  line  is  curved.  It 
is,  indeed,  probable  that  neither  of  the  loga- 
rithmic plots  involving  depth  is  straight;  yet 
there  are  cogent  practical  reasons  for  assuming 
one  or  the  other  to  be  so.  One  reason  is  the 
very  great  convenience  of  the  straight-line  func- 
tion, and  another  that  the  relatively  small  range 
of  the  depth  data  renders  impracticable  such  a 
discussion  of  the  curvature  of  logarithmic  loci 
as  was  made  in  the  case  of  capacity  versus 
slope.  Accordingly,  the  most  orderly  plots  of 
log  d  =fL  (log  S)  and  log  d  =/re  (log  C)  were  com- 
pared with  special  reference  to  curvature.  For 
the  function  of  log  C  the  plots  were  found  to 
indicate  curvature  in  one  direction  only,  while 
for  the  function  of  log  S  they  indicated  slight 
curvatures  in  both  directions,  with  the  straight 
line  as  an  approximate  mean.  The  function 


log  d  =/!  (log  S)  was  accordingly  selected  for  the 
adjustment  of  the  depth  observations;  and  the 
representative  line  on  the  logarithmic  plot  was 
assumed  to  be  straight.  In  accordance  with 
that  assumption,  the  adopted  formula  of  inter- 
polation was 

*-£-- -<21) 


with  its  logarithmic  equivalent, 

log  d  =  log  b'  —  nt  log  S. 


.(22) 


The  coefficient  6'  is  a  depth,  the  depth  cor- 
responding to  a  slope  of  1  per  cent. 

The  data  were  all  plotted  on  logarithmic 
section  paper.  The  notation  was  made  to 
distinguish  depth  measurements  made  at  a 
single  point  by  means  of  the  gage  (see  p.  25) 
from  those  based  on  full  profiles  of  water  sur- 
face and  bed  of  debris.  The  former  were  used 
exclusively  in  the  drawing  of  the  representative 
lines,  but  not  because  they  were  regarded  as 
of  higher  authority.  It  was  thought  best  not 
to  combine  data  which  in  certain  cases  were 
known  to  be  incongruous ;  and  the  gage  observa- 
tions covered  the  whole  range  of  the  work, 
while  the  profiles  did  not.  The  measurements 
by  profile  were  used  in  criticising  the  measure- 
ments by  gage,  and  they  determined  the  accept- 
ance or  rejection  of  certain  gage  measurements. 
It  was  noted  that  in  some  series  of  observa- 
tions the  depth  measurements  by  the  two 
methods  were  in  close  accord,  while  in  others 
there  was  a  large  systematic  difference;  and 
certain  series  were  rejected  because  of  such 
large  differences. 

The  plots  were  made  to  distinguish  also  the 
observations  associated  with  different  modes 
of  traction — the  dune,  smooth,  and  antidune 
modes.  The  observations  with  the  smooth 
mode  were  assumed  to  be  best,  as  a  class;  and 
these,  together  with  the  observations  connected 
with  the  transitional  phases  of  traction,  were 
used  to  fix  an  initial  point  of  each  representa- 
tive line.  The  direction  of  the  line  was  then 
adjusted  by  eye  estimate  to  make  it  repre- 
sentative of  all  the  points  of  the  particular 
series.  In  this  adjustment  consideration  was 
given  to  the  conditions  affecting  the  measure- 
ments of  both  depths  and  slopes. 

The  lines  were  first  drawn  for  those  obser- 
vational series  which  appeared  from  the  plots 
to  be  most  harmonious,  and  for  these  there  was 


ADJUSTMENT   OF   OBSERVATIONS. 


89 


discovered  a  tendency  toward  parallelism, 
within  each  group,  of  lines  associated  with  a 
particular  sand  grade  and  channel  width  but 
differing  as  to  discharge.  For  numerous  other 
groups,  involving  very  irregular  observational 
positions,  such  parallelism  was  assumed;  and 
under  that  assumption  the  bettor  series  of  a 
group  were  made  to  control  the  directions  of 
the  lines  for  the  poorer.  For  three  groups  the 
directions  were  interpolated  by  use  of  the 
directions  found  in  affiliated  groups. 

For  each  observational  series  the  direction 
of  the  representative  line  gave  the  value  of  n^ 
in  equations  (21)  and  (22),  and  the  intersection 
of  the  line  with  the  axis  of  log  d  gave  the  value 
of  I'. 

TABLE  13. —  Value*  ofnt  in  d——^. 


Grade 
of 
debris. 

Value  of  «i  for  trough  having  width  (in  feet)  of— 

0.23 

0.44 

0.66 

1.00 

1.32 

1.96 

i? 

<C 

<<i 

((oF 

(H 

0.44 

!:tt] 

0.50 
.70 

.48-.  58 
.40-.  59 

0.60 
.55 

.48 

0.49 

0.40 
.56 
.40 

0.34 

.53 



[.351 
.30 

.34 
.26 
.30 

-V.»j- 



.28 
.34 

In  Table  13  the  determinations  of  nt  are 
assembled,  being  arranged  vertically  with  re- 
spect to  grade  of  d6bris  and  horizontally  with 
respect  to  width  of  trough.  Collectively  they 
indicate  that,  for  constant  discharge,  the  depth 
varies  inversely  with  the  slope  and  less  rapidly 
than  the  slope.  Comparatively  they  indicate 
considerable  range  in  the  rate  of  variation. 


TABLE  14. — Adjusted  values  of  depth  of  current  (d),  with  values  of  mean  velocity  (  Vm)  and  form  ratio  ^R=—J . 
[»i.  Exponent  in  adjusting  formula  (21);  p.  e.,  probable  error,  in  per  cent,  of  adjusted  values.] 


There  is  some  suggestion  of  system  in  the  dis- 
tribution of  values  of  nlt  and  it  is  especially 
probable  that  to  the  coarser  debris  belong  the 
smaller  values,  but  irregularity  is,  on  the  whole, 
more  in  evidence  than  system.  The  irregu- 
larity is  to  be  ascribed  largely  to  the  imperfec- 
tion of  the  data,  and  imperfection  is  believed 
to  inhere  especially  in  the  measurements  of 
depths  by  means  of  the  gage.  (See  p.  25.) 

Table  14  contains  series  of  adjusted  or  in- 
terpolated values  of  depth.  These  were  com- 
puted for  slopes  with  the  constant  interval  of 
0.2  per  cent,  the  range  of  slope  within  each  se- 
ries being  approximately  the  same  as  the  range 
through  which  the  depth  observations  were 
made.  The  range  is  usually  less  than  the 
range  of  slopes  for  which  capacities  were  com- 
puted, the  difference  being  occasioned  by  the 
fact  that  satisfactory  measurements  of  depth 
could  not  be  made  during  vigorous  antidune 
traction.  The  computations  were  graphic,  the 
values  of  adjusted  depth  being  read  directly 
from  the  logarithmic  plot. 

The  number  of  series  represented  in  the  table 
is  73,  being  one-fourth  less  than  the  number  for 
which  capacities  and  slopes  were  tabulated. 
For  18  series  the  depth  measurements  were  not 
reduced,  the  chief  reasons  for  rejections  being 
either  (1)  that  the  number  of  observations  was 
very  small,  (2)  that  the  observations  included 
none  made  in  connection  with  the  smooth 
phase  of  traction,  and  (3)  that  the  measure- 
ments with  gage  differed  systematically  by  a 
large  amount  from  corresponding  measure- 
ments by  the  method  of  profiles. 


Grade 
w  
Q.... 

nt  
p.  e.. 

(A) 
1.32 
.363 
.50 

1.5 

(A) 
1.32 
.734 
.50 
1.8 

(A) 
1.96 
.363 
.60 

0.7 

(A) 
1.96 
.734 
.60 

1.7 

(A) 
1.96 
1.119 
.60 
3.4 

S 

d 

r« 

R 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

1  25 

0  165 

0.255 

1.47 

0.130 

0.232 

3.35 

0.118 

.4 

.6 

.154 
.126 

1.78 
2.09 

.117 
.096 

0.227 
.185 

2.45 
3.00 

0.172 
.141 

0.136 
.107 

1.36 
1.74 

0.170 
.155 

.167 
.131 
110 

2.24 
2.88 
3  41 

.085 
.067 
056 

.182 
.153 

4.25 
5.05 

.093 
.078 

.8 
1.0 

.098 

2.83 

.074 

.079 

2.35 

!040 

1.2 

.090 

3.10 

.068 

.071 
064 

2.02 
2.87 

.036 
.033 

90 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


TABLE  14. — Adjusted  values  of  depth  of  current  (d),  with  values  of  mean  velocity  (T'm)  and  form  ratio  (  R  =  ~\ — Con. 


Grade.  . 

9 

.093 
.49 
3.3 

(B) 
.23 

.182 
.49 

(B) 
.44 

.093 

fi" 

CB) 
.44 

.182 

&« 

fB) 

.66 
.093 
.55 
2.3 

Q 

p.e.... 

S 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

d               Vm 

R 

0.4 
.6 
.8 
1.0 
1.2 
1.4 
1.6 
l.g 
2.0 
2.2 
2.4 
2.6 

0.128 
.113 
.102 
.094 
.088 
.083 
.079 
.075 
.072 
.069 
.067 

1.73 
1.96 
2.08 
2.36 
2.53 
2.67 
2.82 
2.95 
3.08 
3.20 
3.32 

0.294 
.259 
.235 
.216 
.202 
.190 
.182 
.173 
.167 
.160 
.154 

0.  220 
.184 
.163 
.146 
.135 

1.90 
2.27 
2.56 
2.81 
3.04 

0.502 
.421 
.371 
.336 
.310 

0.200 
.179 
.163 
.150 
.141 
.133 

2.02 
2.17 
2.49 
2.58 
2.86 
3.04 

0.880 
.780 
.715 
.660 
.620 
.583 

0.327 
.292 
.266 
.247 
.231 
.218 

2.43 
2.70 
2.96 
3.02 
3.42 
3.61 

1.43 
1.28 
1.17 
1.08 
1.01 
.96 

0.079             1.80 
.069            2.03 
.063            2.25 
.057             2.45 
.053             2.65 
.050             2.82 
.047             3.00 
.045             3.15 

0.118 
.105 
.095 
.087 
.081 
.076 
.072 
.068 

Grade.  . 

% 

.182 
.40 
2.1 

(B) 
.66 
.363 
.40 
2.1 

(B) 
.66 
.545 
.40 
2.0 

(B) 
1.00 
.182 
.34 
3.3 

(B) 
1.00 
.363 
.34 
2.5 

Q  

p.e.... 

5 

d 

Vm 

R 

d 

Vm            R            d 

Vm 

R 

d 

Va 

R 

d 

Vm 

R 

0.2 
.4 
.6 
.8 
1.0 
1.2 
1.4 
1.6 
1.8 
2.0 
2.2 
2.4 
2.6 

0.208 
.176 
.133 
.118 
.108 
.100 
.095 
.090 
.085 
.082 
.079 

1.33 

1.76 
2.08 
2.34 
2.56 
2.75 
2.94 
3.10 
3.25 
3.38 
3.50 

0.317 
.240 
.203 
.181 
.165 
.153 
.144 
.137 
.130 
.125 
.120 

.  ...              0.42 

- 
i 
4 
I 

i 
7 
4 

1.97 
2.60 
3.04 
3.40 
3.70 
3.97 
4.24 

0.650 
.490 
.418 
.370 
.340 
.315 
.296 

0.239 
.188 
.164 
.148 
.137 
.128 
.122 
.117 
.112 
.108 
.105 

1.53 
1.94 
2.24 
2.46 
2.66 
2.83 
2.98 
3.12 
3.25 
3.36 
3.48 

0.239 
.188 
.164 
.148 
.137 
.128 
.122 
.117 
.112 
.108 
.105 

0.246 
.210 
.187 
.171 
.158 
.148 

2.22       0.373         .32 
2.63         .318         .27 
2.95        .283         .24 
3.23         .259         .22 
3.49         .240        .20 
3.71         .226         .19 

0.101 
.091 
.084 
.079 
.075 
.072 
.069 
.066 
.064 
.062 
.061 

1.81 
2.00 
2.15 
2.39 
2.40 
2.51 
2.62 
2.71 
2.80 
2.92 
3.00 

0.101 
.091 
.084 
.079 
.075 
.072 
.069 
.066 
.064 
.062 
.061 

Grade.  . 

(B) 
1.00 
.734 
.34 
1.4 

(B) 
1.32 
.182 
.70 
2.5 

(B) 
1.32 
.545 
.70 
2.4 

(B)                                     (B) 
1.32                                       1.96 
.734                                       .363 
.70                                         .55 
2.6                                        3.3 

<J  ..  . 

tii  

p.e.... 

S 

d 

Vm 

R 

d 

Vm 

R 

1 

Vm 

R 

d            Vm          R            d 

Vm 

R 

0.2 
.4 

.6 
.8 
1.0 

:.2 

1.4 
1.6 
1.8 
2.0 
2.2 
2.4 

0.347 
.275 
.239 
.217 
.200 

2.10 
2.68 
3.10 
3.41 
3.67 

0.347 
.275 
.239 
.217 
.200 

0.200 
.124 
.094 
.077 
.006 
.058 
.052 
.047 
.043 
.040 
.038 
.036 

0.68 
1.12 
1.48 
1.82 
2.13 
2.42 
2.70 
2.96 
3.20 
3.45 
3.70 
3.92 

0.153 
.094 
.071 
.057 
.050 
.046 
.039 
.036 
.033 
.031 
.029 
.027 

0.233 
.175 
.143 
.123 
.108 
.097 

1.76 
2.35 
2.88 
3.38 
3.82 
4.27 

0.176 
.133 
.108 
.093 
.082 
.073 

0.256        2.10       0.194       0.117 
.193         2.83         .146         .094 
.158        3.48         .120         .080 

.                       .1*71 

1.58 
1.98 
2.33 
2.65 
2.93 
3.20 
3.43 
3.67 
3.88 

0.059 
.047 
.041 
.036 
.033 
.030 
.028 
.026 
.025 

064 

059 

054 

051 

.048 

Grade.  . 

(B) 

1.96 
.545 
.55 
2.8 

(B) 
1.96 
.734 
.55 
5.4 

(B) 
1.96 
1.119 
.55 

3.8 

(C) 
.44 
.093 
.44 
0.6 

% 

.182 
.44 

0.7 

p.e.... 

S 

d               Vm 

S             d            Vm           R 

d 

Vm 

R 

d 

Vm 

R             d              Vm 

R 

0.2 
.4 
.6 
.8 
1.0 
1.2 
1.4 
1.6 
1.8 

n  2* 

«         1.40      0.136 
13        2.05         .094 
7         2.55         .075 
«         2.98         .064 
2        3.35         .057 

0.4W 
.27 
.21" 
.18. 
.IK 
.14' 

)          1.45 
2.13 
2.66 
>        3.11 
1        3.51 
3.89 

0.203 
.138 
.110 
.094 
.083 
.075 

0.108         1.63 
.135        2.04 
.115        2.38 
.102        2.70 
.092         2.30 
.085        2.26 

(I 

086         .li 
069         .14 
059         .11 
052         .11 
047 

0.163 
.143 
.130 
.120 
.112 
.106 
.101 

.28 
.46 
.62 
.75 
.87 
2.00 
2.10 

0.370         0.240             1.73 
.328            .213             1.96 
.298           .192             2.16 
.275           .177             2.34 
.256           .165             2.50 
.  242           .  157             2.  66 
230 

0.546 
.480 
.436 
.400 
.375 
.355 

043 

I    - 

ADJUSTMENT   OF   OBSERVATIONS. 


91 


TABLE  14. — Adjusted  values  of  depth  of  current  (d),  with  values  of  mean  velocity  (  Vm]  and  form  ratio  \^=^1) — Con. 


Grade.. 

(C) 
.66 

(C) 
.66 

(C) 
.66 

(C) 
.66 

a 

Q...     . 

.093 

.182 

.363 

.545 

.734 

»i  
p.e.... 

.56 
0.8 

.56 
1.8 

.56 
1.4 

.56 
1.3 

.56 
1.9 

S 

| 

Vm 

R 

d 

Vm 

R 

<t 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

0  2 

0  270 

1  03 

0.410 

0.562 

1.43 

0.850 

0  199 

183 

1  52 

277 

0  288 

1  88 

0.438 

381 

2  13 

580 

.6 
.8 
1.0 
LI 

.105 
.090 
.079 
.071 

.35 

.58 
.79 
.97 
2  14 

.160 
.135 
.119 
.107 
099 

.146 
.125 
.110 
.100 
092 

1.89 
2.21 
2.50 
2.77 
3  00 

.222 
.188 
.167 
.150 
138 

.230 
.196 
.173 
.157 

2.38 
2.80 
3.20 
3.53 

.350 
.298 
.264 
.238 

.304 
.2GO 
.228 
.206 

2.68 
3.16 
3.60 
4.00 

.461 
.392 
.346 
.312 

0.340 
.290 
.255 
.230 

3.26 
3.82 
4.34 
4.80 

0.515 
.440 
.386 
.349 

1  6 

OG1 

2  30 

092 

085 

3  23 

.128 

2  45 

086 

080 

3  44 

120 

2  0 

054 

2  GO 

082 

075 

3  64 

.113 

077 

071 

3  83 

107 

2  4 

2  86 

074 

2  6 

046 

3  00 

071 

Grade.  . 

(C) 
1.00 

(C) 
1.00 

(C) 
1.00 

(C) 
1.00 

(C) 
1.32 

Q... 

.182 

.363 

.545 

.734 

.182 

.53 

.53 

.53 

.53 

.58 

p.e.... 

1.9 

0.9 

1.5 

2.2 

1.7 

S 

d 

Vn 

R 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

0  2 

0  307 

1  18 

0  307 

0.450 

1.66 

0.450 

212 

1  72 

212 

310 

2  39 

.310 

g 

113 

1  62 

113 

171 

2  14 

171 

.250 

2.96 

.250 

0.100 

1.38 

0  075 

.8 
.0 
.2 

.098 
.086 
.078 

1.88 
2.13 

2;  34 

2  55 

.098 
.086 
.078 

.147 
.130 
.118 
108 

2.49 
2.80 
3.08 
3  34 

.147 
.130 
.118 
108 

0.190 
.170 
.153 
141 

2.86 
3.22 
3.54 
3  84 

0.190 
.170 
.153 
.141 

.214 
.190 
.172 

3.45 
3.88 
4.26 

.214 
.190 
.172 

.084 
.074 
.067 
.061 

1.65 

1.88 
2.09 
2.29 

.064 
.056 
.050 
.046 

6 

067 

2  74 

067 

102 

3  60 

102 

.132 

4  14 

.132 

.056 

2.47 

.043 

063 

123 

4  52 

123 

.053 

2  73 

040 

2  0 

.059 

3  08 

.059 

.050 

2.81 

.037 

2  2 

056 

3  24 

056 

.047 

2.% 

.035 

2  4 

054 

3  38 

.054 

.045 

3.12 

.034 

2  6 

.043 

3.27 

.032 

Grade.  . 

(C) 
1  32 

L? 

(C) 
1.32 

(C) 
1.96 

(C) 
1.96 

Q  

.363 

.545 

.734 

.363 

.545 

48 

.48 

.48 

.48 

.48 

p.e.... 

2.3 

.8 

1.1 

1.2 

1.1 

S 

1 

Vm 

R 

I 

r« 

R 

d 

Vm 

R 

d 

Vm 

R 

1 

Vm 

R 

0  2 

0  344 

1  63 

0.261 

.4 

0.168 

1.66 

0.128 

.246 

2.27 

.187 

0.132 

1.18 

0.068 

0.166 

1.66 

0.085 

.6 
.8 
1.0 
1.2 
1.4 
1.6 
1  8 

.138 
.120 
.108 
.099 
.092 
.086 

2.00 
2.28 
2.54 
2.76 
2.96 
3.15 

.105 
.092 
.082 
.075 
.070 
.065 

0.174 
.152 
.136 
.125 
.116 
.108 

2.36 
2.71 
3.02 
3.30 
3.54 
3.78 

0.132 
.115 
.104 
.095 
.088 
.083 

.203 
.177 
.158 
.145 
.135 
.124 

2.76 
3.17 
3.50 
3.84 
4.14 
4.40 

.153 
.134 
.120 
.110 
.102 
.096 

.108 
.095 
.085 
.078 
.072 
.068 
.064 

1.44 
1.66 
1.85 
2.02 
2.18 
2.32 
2.46 

.056 
.048 
.043 
.040 
.037 
.035 
.033 

.136 
.118 
.106 
.097 
.090 
.084 
.079 

2.02 
2.34 
2.61 
2.86 
3.10 
3.30 
3.50 

.069 
.060 
.054 
.049 
.046 
.043 
.040 

2  0 

.061 

2.59 

.031 

.075 

3.69 

.038 

2  2 

.058 

2.71 

.030 

Grade.. 

(C) 
1  96 

(C) 
1.96 

(D) 

O.tifi 

(D) 

0.06 

(D) 
0.66 

Q  

.734 

1.119 

.093 

.182 

.545 

48 

.48 

.40 

.40 

.40 

p.e.... 

1.0 

1.0 

1.3 

1.1 

1.5 

8 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

0  2 

0.461 

1.81 

0.640 

4 

0  194 

1  96 

0  099 

0  240 

2  40 

0  122 

0.172 

1.61 

0.258 

.349 

2.38 

.531 

| 

2  38 

081 

197 

2  92 

100 

.143 

1.92 

.217 

.2% 

2.80 

.450 

.8 
1.0 
1.2 
1  4 

.138 
.124 
.113 

105 

2.73 
3.04 
3.31 
3  56 

.070 
.063 
.058 
053 

.172 
.153 
.140 

3.36 
3.73 
4.08 

.088 
.078 
.072 

0.087 
.080 
.074 
.069 

1.63 
1.78 
1.91 
2.02 

0.132 
.132 
.112 
.105 

.127 
.115 
.107 
.100 

2.17 
2.40 
2.59 
2.77 

.192 
.174 
.161 
.151 

.265 
.242 
.224 
.212 

3.14 
3.42 
3.69 
3.91 

.403 
.369 
.342 
.322 

1  6 

099 

3  80 

.050 

.066 

2.13 

.100 

.094 

2.92 

.143 

.200 

4.13 

.305 

1.8 

.003 

2.23 

.093 

.090 

3.07 

-     .135 

.191 

4.33 

.291 

2  0 

.060 

2.33 

.091 

.086 

3.21 

.130 

.183 

4.50 

.280 

058 

2  42 

.088 

.082 

3.36 

.125 

2  4 

056 

2.50 

.085 

.079 

3.48 

.120 

054 

2  58 

.082 

.077 

3.60 

.116 

92 

TABLE  14. 


TRANSPORTATION    OF    DEBRIS    BY    RUNNING    WATER. 
— Adjusted  values  of  depth  of  current  (d),  with  values  of  mean  velocity  (Vm~)  and  form  ratio  \R=  ,) — Con. 


Grade.  . 

(D) 
1.00 

(D) 
1.00 

(D) 
1.00 

(D) 
1.00 

<r» 

1.32 

Q 

.182 

.363 

.545 

.734 

.363 

.42 

.42 

.42 

.42 

.40 

p.e.... 

.8 

1.8 

.9 

1.5 

2.1 

S 

t 

Vm 

R 

d 

Vm 

R 

d 

vm 

R 

d 

Vm 

R 

d 

Vm 

R 

0  2 

0.278 

1.32 

0.278 

207 

1  77 

.207 

0  252 

2  16 

0.252 

0  163 

1  68 

.6 
.8 
1.0 
1.2 
1.4 
1.6 
1.8 
2.0 
2  2 

0.116 
.103 
.094 
.087 
.081 
.077 
.073 
.070 
067 

1.57 
1.76 
1.93 
2.09 
2.22 
2.34 
2.47 
2.57 
2  07 

6.  iie 

.103 
.094 
.087 
.081 
.077 
.073 
.070 
.067 

.174 
.154 
.140 
.130 
.122 
.115 
.108 
.104 
.101 

2.10 
2.36 
2.60 
2.81 
3.00 
3.18 
3.33 
3.48 
3.63 

.174 
.154 
.140 
.130 
.122 
.115 
.108 
.104 
.101 

.213 
.188 
.172 
.158 
.149 
.141 
.134 
.128 
.123 

2.56 
2.89 
3.19 
3.43 
3.67 
3.88 
4.08 
4.27 
4.44 

.213 
.188 
.172 
.158 
.149 
.141 
.134 
.128 
.123 

0.260 
.231 
.210 
.195 
.183 
.173 
.164 
.157 

2.87 
3.21 
3.51 
3.80 
4.04 
4.28 
4.49 
4.69 

0.200 
.231 
.210 
.195 
.183 
.173 
.164 
.157 

.138 
.124 
.113 
.105 
.099 
.094 
.090 
.086 

1.98 
2.22 
2.43 
2.62 
2.78 
2.93 
3.07 
3.20 

.106 
.094 
.086 
.080 
.075 
.071 
.068 
.065 

2  4 

.065 

2.77 

.065 

.097 

3.77 

.097 

2.6 

.003 

2.87 

.063 

2  8 

061 

2  95 

.061 

3  0 

.059 

3.04 

.059 

Grade.  . 

(D) 
1.32 

(E) 
.66 

(E) 
.66 

(E) 
1.00 

(E) 
1.00 

Q 

.734 

.363 

.734 

.182 

.363 

.59 

[•35] 

[.35] 

.34 

.34 

p.e.... 

1.7 

1.6 

2.6 

1.6 

S 

t 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

0  2 

0  423 

1.30 

0.323 

0  252 

1  44 

0  252 

281 

1  97 

.214 

198 

1  83 

5 

222 

2  50 

.169 

173 

2  10 

173 

g 

.187 

2.97 

.143 

.157 

2  32 

157 

.0 
.2 
4 

.164 
.147 

3.38 
3.78 

.125 
.112 

0.129 
.122 
115 

2.14 

2.28 
2  40 

0.195 
.183 
173 

0.206 
.194 
183 

2.66 
2.83 
3  00 

0.313 
.293 

285 

0.097 
.091 
087 

1.88 
2.02 
2  13 

0.097 
.091 
087 

.146 
.137 

129 

2.50 
2.66 
2  80 

.146 
.137 
129 

6 

.109 

2.52 

.166 

.175 

3.14 

.265 

.083 

2.22 

083 

123 

2  94 

123 

g 

080 

2  31 

080 

118 

3  05 

118 

2  0 

.077 

2  39 

077 

114 

3  16 

114 

2  2 

.112 

3  27 

112 

2  4 

108 

3  37 

108 

Grade.  . 

(E) 
1.00 

(F) 
.66 

(F) 
.66 

(F) 
1.00 

(F) 
1  00 

a 

.734 

.182 

363 

182 

363 

.34 

.30 

.30 

.26 

26 

p.e.... 

2.6 

1.5 

0.2 

1.2 

S 

i 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

a 

vm 

R 

d 

Vm 

R 

0.2 

0.395 

1.85 

0.395 

.4 

.312 

2.34 

.312 

.6 

.272 

2.70 

.272 

.8 

.246 

2.98 

.246 

0.227 

2.43 

0.345 

0  162 

2  24 

0  162 

1.0 
1.2 
1.4 
1.6 
1  8 

.228 
.214 
.203 
.194 

3.23 
3.44 
3.63 
3.80 

.228 
.214 
.203 
.194 

0.134 
.127 
.122 
.117 
113 

2.05 
2.16 
2.28 
2.36 
2  46 

0.205 
.194 
.185 
.178 
172 

.213 
.201 
.192 
.185 
178 

2.59 
2.74 
2.86 
2.98 
3  08 

.321 
.303 
.290 
.280 
270 

0.099 
.095 
.091 
.088 
085 

1.83 
1.92 
2.00 
2.07 
2  14 

0.099 
.095 
.091 
.088 
085 

.153 
.145 
.139 
.135 
131 

2.38 
2.50 
2.60 
2.69 
2  78 

.153 
.145 
.139 
.135 
131 

2.0 

.108 

2.52 

.167 

.173 

3  18 

262 

083 

2  20 

083 

127 

2  85 

127 

2.2 

.106 

2.60 

.162 

.167 

3.28 

.254 

.081 

2  26 

081 

2.4 

.103 

2.67 

.157 

079 

2  32 

079 

2.6 

.101 

2.74 

.153 

.078 

2.37 

078 

ADJUSTMENT   OF   OBSERVATIONS. 


93 


TABLE  14. — Adjusted  values  of  depth  of  current  (d),  with  values  of  mean  velocity  ( Vm),  and  form  ratio  (  R=—J — Con. 


Grade, 
w  
Q  
nr.  — 

p.e... 

(F) 
1.32 
.363 

I? 

(G) 
.66 
.363 
[.26] 
0.2 

(G)                                             (G) 
.66                                             .66 
.734                                          1.119 
1.26]                                              [.26] 
0.2                                             0.4 

(G) 
1.00 
.363 
.30 
0.4 

s 

d             Vm            R 

d            Vm          K 

d 

Ira  R  d  Vm  R 

d 

vm 

R 

0.6 
.8 
1.0 
1.2 
1.4 
1.6 
1.8 
2.0 
2.2 
2.4 
2.6 
2.8 
3.0 

0.389 
.358 
.338 
.320 
.306 
.294 
.285 
.276 
.270 
.263 
.257 
.252 

2.86  0.588  0.465  3.68  0.705 
3.10  .540  .436  3.92  .660 
3.30  .508  .413  4.12  .628 
3.48  .481  .396  4.30  .600 
3.63  .  INI  .382  4.44  .580 
3.78  .443  .371  4.58  .561 
3.90  .430  .361  4.70  .548 
4.02  .418  .352  4.80  .532 
4.13  .405  .346  4.90  .522 
4.22  .395  .338  5.00  .512 
4.32  .387  .332  5.10  .502 
4.41  .378  .327  5.19  .494 

0.  222         2.  49       0.  336 
.218         2.65         .315 
.198         2.80         .300 
.189         2.91          .287 
.183         3.02         .277 
.  176        3.  13         .268 
.172         3.23         .260 
.167         3.30         .253 
.163         3.39         .247 
.158         3.46         .242 

0.127         2.17       0.096 
.121         2.27          .092 
.116         2.36         .088 
.112        2.44         .085 
.108        2.52         .083 
.106        2.59         .081 
.103        2.65         .079 

0.158 
.149 
.143 
.137 
.132 
.128 
.124 
.121 
.118 
.116 
.113 

2.30 
2.42 
2.54 
2.65 
2.74 
2.83 
2.93 
3.00 
3.07 
3.14 
3.22 

0.158 
.149 
.143 
.137 
.132 
.128 
.124 
.121 
.118 
.116 
.113 

1 

Grade.. 

(G) 
1.00 
.734 
.30 
0.6 

(G) 
1.00 
1.119 
.30 
0.5 

(G) 
1.32 
.363 
.26 
1.1 

Si 

.734 
.26 
1.2 

Q  

p.  e  

S 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

d 

Vm 

R 

0.6 
.8 
1.0 
1.2 
1.4 
1.6 
1.8 
2.0 
2.2 
2.4 
2.6 
2.8 
3.0 
3.2 

0.287 
.363 
.246 
.232 
.222 
.213 
.206 
.199 
.194 
.188 
.184 
.180 

2.  57 
2.80 
2.99 
3.16 
3.30 
3.43 
3.56 
3.68 
3.78 
3.87 
3.96 
4.05 

0.287 
.263 
.246 
.232 
.222 
.213 
.206 
.199 
.194 
.188 
.184 
.180 

0.376 
.344 
.322 
.302 
.290 
.279 
.269 
.260 
.253 
.247 
.241 

3.00 
3.27 
3.50 
3.70 
3.86 
4.04 
4.18 
4.31 
4.43 
4.55 
4.61 

0.376 
.344 
.322 
.302 
.290 
.279 
.289 
.260 
.253 
.247 
.241 

0.216 
.203 
.194 
.186 
.180 
.174 
.169 
.165 
.162 
.158 

2.60 
2.75 
2.89 
3.00 
3.12 
3.22 
3.30 
3.39 
3.47 
3.54 
3.61 

0.164 
.154 
.147 
.141 
.136 
.132 
.128 
.125 
.122 
.120 
.118 

0.114 
.112 
.108 
.105 
.103 
.101 
.099 
.098 
.0% 

2.41 

2.48 
2.55 
2.62 
2.67 
2.74 
2.78 
2.83 
2.87 

0.087 
.084 
.082 
.080 
.078 
.077 
.075 
.074 
.073 

Grade.. 

(0) 
1.32 
1.119 
.26 

0.7 

(H) 
.66 
.363 
.34 

1.7 

(H) 
.66 
.734 
.34 
0.9 

(H) 
.66 
1.119 
.34 
0.5 

Q    

S 

i 

Vm 

R 

d 

Vm 

* 

• 

Ym 

R 

t 

Ym 

R 

0.6 
.8 
1.0 
1.2 
1.4 
1.6 
1.8 
2.0 
2.2 
2.4 
2.6 
2.8 
3.0 

0.303 
.281 
.264 
.251 
.241 
.232 
.226 
.219 
.214 
.208 
.203 

2.80 
3.03 
3.22 
3.38 
3.52 
3.66 
3.78 
3.88 
3.98 
4.07 
4.17 

0.230 
.213 
.200 
.190 
.183 
.176 
.170 
.165 
.162 
.157 
.154 

0.560 
.507 
.470 
.441 
.420 
.400 
.385 
.372 
.360 
.349 
.341 

2.83 
3.12 
3.36 
3.59 
3.78 
3.94 
4.10 
4.25 
4.40 
4.50 
4.64 

0.850 
.770 
.713 
.670 
.636 
.608 
.584 
.564 
.548 
.530 
.518 

0.382 
.354 
.333 
.316 
.302 
.290 
.280 
.271 
.263 
.256 
.250 
.244 

2.93 
3.16 
3.36 
3.53 
3.70 
3.84 
4.00 
4.13 
4.25 
4.36 
4.46 
4.57 

0.580 
.538 
.504 
.479 
.458 
.440 
.425 
.411 
.399 
.388 
.379 
.370 

0.220 
.207 
.197 
.188 
.181 
.174 
.168 
.164 
.159 

2.52 
2.68 
2.82 
2.96 
3.07 
3.18 
3.29 
3.39 
3.48 

0.336 
.316 
.300 
.286 
.276 
.267 
.258 
.250 
.243 

94 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


PRECISION. 

Probable  errors  were  computed  in  the  manner 
described  on  page  73.  The  largest  error 
found  for  a  series  of  adjusted  values  of  depth 
is  ±5.5  per  cent;  and  the  arithmetical  mean 
of  the  66  determinations  is  ±1.70  per  cent. 
The  mean  of  66  determinations  of  the  probable 
error  of  an  observation  of  depth  is  ±5.47  per 
cent;  and  the  arithmetical  mean  of  the  (710) 
residuals,  or  differences  (irrespective  of  sign) 
between  observed  and  adjusted  depths,  is 
6.65  per  cent. 

The  residuals  of  any  series,  computed  as 
fractional  parts  of  the  quantity  measured,  are 
greater  for  small  depths  than  for  large.  Com- 
puted in  fractions  of  a  foot,  they  do  not  vary 
notably  with  depth. 

The  average  value  of  the  depths  to  which 
these  measures  of  precision  pertain  is  0.176 
foot.  Using  this  factor  to  convert  the  pre- 
ceding average  percentage  errors  into  (ap- 
proximately) equivalent  linear  errors,  we  have, 
for  the  average  of  probable  errors  of  adjusted 
depths  ±0.003  foot;  for  the  average  of  probable 
errors  of  observed  depths  ±0.010  foot,  and 
for  the  average  residual  0.011  foot. 

The  errors  thus  estimated  include  (1)  the 
strictly  accidental  errors  of  observation,  (2)  the 
more  or  less  systematic  influences  exerted  on 
the  measurements  by  the  diverse  modes  of 
traction,  and  (3)  errors  occasioned  by  the  as- 
sumptions underlying  the  method  of  reduc- 
tion; but  they  do  not  include  such  systematic 
errors  as  are  shown  by  comparing  gage  meas- 
urements with  profile  measurements.  (See  p. 
26.)  In  connection  with  the  descriptions  of 
methods  of  measurement  it  is  stated  that  the 
probable  error  of  a  single  measurement  by  gage 
was  computed  from  the  comparison  of  values 
by  two  methods  (in  all  cases  where  both  are 
used),  and  on  the  assumption  that  the  measures 
by  the  profile  method  are  relatively  accurate. 
This  computation  gave  a  value  smaller  than 
±  0.010  foot,  namely,  ±  0.007  foot.  The  groups 
of  measurements  to  which  the  two  estimates  of 
precision  apply  are  not  the  same,  although  they 
overlap.  One  group  includes  all  gage  measure- 
ments of  66  series,  their  number  being  710;  the 
other  group  includes  all  those  gage  measure- 
ments of  the  92  series  which  were  checked  by 
profile  measurements,  their  number  being  118. 


By  comparing  these  results  with  those  re- 
ported on  page  74,  it  is  seen  that  the  computed 
probable  errors  for  depth  are  smaller  than  those 
for  capacity.  It  is  nevertheless  believed  that 
the  measurements  of  load  were  more  precise 
than  those  of  depth.  The  discrepancy  is  ac- 
counted for  by  considering,  first,  that  the  esti- 
mates of  precision,  instead  of  applying  simply  to 
the  measurements  of  load  and  depth,  apply  to 
capacity  for  load  as  a  function  of  slope  and  to 
depth  as  a  function  of  slope;  and,  second,  that 
the  relation  of  load  to  slope 'is  subject  to  con- 
tinual rhythmic  variation,  while  the  relation  of 
depth  to  slope  is  little  influenced  by  that  varia- 
tion. 

MEAN    VELOCITY. 

As  the  discharge,  Q,  equals  the  product  of  the 
sectional  area  of  the  current,  wd,  by  the  mean 
velocity,  Vm,  we  have 

Fra  =  4-  -(23) 


For  each  observational  series,  Q  and  w  are  con- 
stant and 

T7         .1 


.(24) 


Substituting  for  d  its  value  in  the  interpolation 
equation  (21),  and  remembering  that  6'  is  a 
constant  for  each  series,  we  obtain 


.(25) 


By  means  of  (23)  a  value  of  mean  velocity  was 
computed  for  each  adjusted  value  of  d,  and 
these  values  are  given  in  Table  14.  They  in- 
volve all  the  assumptions  of  the  formula  for  the 
reduction  of  the  depth  observations  and  have 
the  same  fractional  measures  of  precision. 

FORM  RATIO. 

The  adjusted  values  of  d  were  used  also  for 
the  computation  of  the  form  ratio,  R,  which  is 
the  quotient  of  the  depth  of  the  current  by  its 
width;  and  a  value  of  R  is  tabulated  with  each 
value  of  d.  Within  each  observational  series 
the  form  ratios  are  proportional  to  the  depths, 
and  they  have  the  same  measures  of  precision. 

GRAPHIC    COMPUTATION. 

All  these  computations  were  made  by  graphic 
methods.  For  each  observational  series  a  plot 


ADJUSTMENT    OF    OBSERVATIONS. 


95 


was  made  on  logarithmic  section  paper.  By 
way  of  illustration  the  plot  for  grade  (B), 
width  1.96  feet  and  discharge  1.119  ft.3/sec.  is 
reproduced  in  figure  28,  the  finer  lines  of  the 
logarithmic  net  being  omitted.  The  vertical 
lines  represent  values  of  slope;  the  horizontal 
lines,  capacity,  depth,  mean  velocity,  and  form 


-C- 


100  Vm 


.1.0 


I 


-.2 


.3 


R, 


.4  .6 

SJope 


.8      1.0 


2.0 


FIGUKE  28.— Logarithmic  computation  sheet,  combining  relations  of 
capacity,  mean  velocity,  form  ratio,  and  slope. 

ratio.  For  the  convenience  of  having  the 
graphs  close  together  their  scales  are  made  to 
differ,  the  ratio  of  one  to  another  being  10,  100, 
or  1,000.  In  the  particular  instance  shown  in 
figure  28  the  same  line  represents  C=  100, 
d  =  0.1,  Vm  =  1.0,  and  5  =  0.1.  With  use  of  this 
notation  were  plotted  the  equations  for  C,  d, 


Vm,  and  R,  as  functions  of  £.  In  their  loga- 
rithmic forms  these  are 

log  d  =  log  &'  —  ttj  log  $ 

log  Fm=log  ^-log  <Z=log  —log  I'  +  n,  log  8 
log  R  =  log  d  —  log  w  =  log  —  —  «•!  log  $ 

log  (7=  log  &t  +  »  log  (S-ff) 

Their  loci  are  the  straight  lines  marked  d,  Vm, 
and  R  and  the  curve  marked  C. 

After  the  preparation  of  these  constructions 
the  values  of  d,  Vm,  and  R,  corresponding  to  the 
selected  series  of  values  of  8,  were  read  from  the 
sheets,  affording  the  data  of  Table  14.  The 
sheets  had  also  many  other  uses,  for  in  record- 
ing the  relations  of  four  interdependent  varia- 
bles to  slope  they  also  recorded  their  relations 
to  one  another.  The  points  of  the  four  loci 
which  lie  in  the  same  vertical  represent  corre- 
sponding values  of  the  several  variables,  and 
this  property  made  it  possible  to  read  from  the 
plot  the  value  of  a  variable  corresponding  to  a 
particular  value  of  one  of  the  others.  For 
example,  if  it  is  desired  to  learn  the  capacity 
corresponding  to  a  mean  velocity  of  2  ft./sec., 
the  intersection  of  the  velocity  line  with  the 
line  representing  2  on  the  scale  of  velocities  is 
first  found.  From  that  intersection  a  vertical 
is  followed  or  drawn  to  the  capacity  line  and 
the  position  of  the  second  intersection  is  read  on 
the  capacity  scale. 

A  large  part  of  the  numerical  data  cited  in 
the  following  discussions  were  either  taken 
directly  from  the  computation  sheets  or  based 
upon  them. 

The  accuracy  of  the  computations  by  loga- 
rithmic graph  may  be  characterized  by  saying 
that  it  is  slightly  below  that  by  slide  rule.  The 
theoretic  accuracy  is  the  same,  but  tests  of  uhe 
logarithmic  paper  employed  showed  it  to  be  a 
less  perfect  instrument  than  the  slide  rule, 
which  was  used  for  a  large  body  of  routine 
computations. 


CHAPTER  III.— RELATION  OF  CAPACITY  TO  SLOPE. 


INTRODUCTION. 

A  series  of  chapters,  beginning  with  this  one, 
are  given  to  the  discussion  of  the  observational 
data.  The  discussions  make  use  of  the  ad- 
justed values  of  capacity  for  stream  traction, 
slope  of  stream  bed,  and  depth  of  current, 
with  their  derivatives,  contained  in  Tables  12 
and  14.  Associated  with  those  adjusted  values 
are  certain  grades  of  transported  material,  or 
degrees  of  fineness,  certain  widths  of  channel, 
and  certain  discharges  of  the  transporting 
stream.  The  leading  subjects  of  discussion  are 
the  relations  of  capacity  to  slope,  discharge, 
fineness,  and  form  ratio,  but  consideration  is 
also  given  to  the  relations  of  capacity  to  depth 
and  velocity,  and  to  the  relations  which  duty, 
efficiency,  and  depth  bear  to  various  condi- 
tions. The  discussion  is  essentially  empiric,  its 
course  being  guided  in  small  degree  only  by 
theoretic  considerations. 

The  treatment  of  the  relation  of  capacity  to 
slope  first  views  it  as  conditioned  by  channels 
of  fixed  width,  and  then  as  subject  to  the  rela- 
tively ideal  condition  of  fixed  form  ratio. 

IN  CHANNELS  OF  FIXED  WIDTH. 

THE    CONDITIONS. 

In  each  observational  series  the  width  of 
channel  was  constant,  and  so  also  were  the 
discharge  and  the  grade  of  debris  constituting 
the  load.  As  the  load  was  changed,  the  slope 
responded;  velocity  responded  to  change  of 
slope;  and  with  variation  of  velocity  went  varia- 
tion of  depth.  The  ratio  of  depth  to  width,  or 
the  form  ratio,  was  therefore  a  variable;  so 
that  the  stream  which  dragged  a  large  load 
down  a  steep  slope  differed  in  form,  and  to  that 
extent  in  type,  from  the  stream  which  moved 
a  small  load  along  a  gentle  slope.  In  a  few 
cases  it  is  possible  so  to  combine  data  from 
different  series  as  to  discover  the  relation  of 
capacity  to  slope  for  streams  which  have  simi- 
lar cross  sections;  and  these  will  be  examined 
in  another  place;  but  the  principal  discussion 
relates  to  streams  with  constant  width  and 
variable  form  ratio. 

96 


THE    SIGMA    FORMULA.1 


Those  properties  of  the  formula 


(10) 


which  determined  its  adoption  for  the  reduction 
of  the  observational  data  to  a  more  orderly 
system  led  also  to  the  consideration  of  its 
availability  as  an  empiric  formula  for  the  gen- 
eral relation  of  the  stream's  capacity  for  trac- 
tion to  the  slope  of  its  bed.  With  a  view  to 
this  second  use,  the  specific  values  of  &1;  <j,  and 
n  derived  for  the  purpose  of  the  reduction  — 
values  which  are  recorded  in  Table  15  —  were 
arranged  and  combined  in  various  ways,  in 
order  to  discover,  if  possible,  definite  relations 
to  the  several  conditions  in  accordance  with 
which  the  experiments  were  varied.  It  has 
already  been  noted  (p.  71)  that  the  critical 
slope,  a,  varies  inversely  with  fineness  of 
debris,  with  discharge,  and  (probably)  with 
range  of  fineness  within  a  grade,  and  that  it 
varies  inversely  with  width  of  trough  when 
that  width  is  relatively  small,  but  directly 
with  width  when  width  is  relatively  large. 


(26) 


written  to  express  these  relations  in  symbols, 
introduces  H  to  designate  range  of  fineness  and 
distinguishes  trend  of  function  by  means  of 
accents.  As  the  notation  by  accents  will  be 
frequently  used,  its  definition  may  be  made 
explicit.  Where  the  function  is  direct,  or 
increasing,  its  value  increasing  with  the  in- 
crease of  the  independent  variable,  the  symbol 
of  the  variable  is  given  the  acute  accent  ('). 
Where  the  function  is  inverse,  or  decreasing,  its 
value  decreasing  with  increase  of  the  variable, 
the  grave  accent  (v)  is  used.  For  a  maximum 
function,  first  increasing  and  then  decreasing 
(A)  is  used,  and  for  a  minimum  function  ("). 
The  discussion  of  the  values  of  bt  showed  (1  ) 
that  they  vary  directly  and  in  marked  degree, 
but  irregularly,  with  F2,  (2)  that  they  vary 

i  Since  these  lines  were  penned  1  have  discovered  that  this  title 
duplicates  one  in  the  field  of  higher  mathematics.  Nevertheless  I 
retain  it  because  of  its  mnemonic  convenience.  The  two  fields  of 
application  are  so  distinct  that  serious  confusion  will  not  be  occasioned. 


RELATION  OF  CAPACITY  TO  SLOPE. 


97 


directly  and  approximately  in  simple  ratio, 
with  Q,  and  (3)  that  they  vary  increasingly,  if 
at  all,  but  very  slightly  with  w,  while  (4)  a  rela- 
tion to  H  could  not  be  disentangled  from  the 
relation  to  F2.  The  discussion  of  the  values  of 
n  showed  (1  )  that  they  vary  inversely  and  irreg- 
ularly with  F2,  (2)  that  they  vary  inversely 
and  more  regularly  with  Q,  and  (3)  that  the 
variation  in  respect  to  w  is  direct  for  small 
widths  and  inverse  for  large  widths,  while  (4) 
the  relation  to  H  is  not  separable. 

The  data  on  the  three  parameters  are  sum- 
marized in  equation  (27),  which  is  an  expan- 
sion of  equation  (10).  In  all  probability  /,  and 
fu  are  as  complex  as/,  but  no  factors  are  intro- 
duced of  which  the  influence  was  not  definitely 
shown  by  the  discussion. 


and  have  purposely  omitted  details.  The 
results,  despite  important  qualifications,  show 
clearly  that  any  general  expression  of  the  law 
connecting  capacity  and  slope  which  might  be 
based  on  formula  (10)  would  be  highly  complex. 
With  reference  to  the  main  subject  of  this 
chapter,  the  following  section  is  of  the  nature 
of  a  digression,  its  purpose  being  to  define  a 
method  and  terminology  used  in  several  of  the 
succeeding  chapters. 

THE     POWER     FUNCTION      AND     THE     INDEX     OF 
RELATIVE    VARIATION. 

One  of  the  algebraic  forms  to  which  the  title 
power  function  is  applied  is 

y  =  axn  ...........  --(28) 


•,*,«      _(27)      If  the  coefficient  be  suppressed,  leaving 


This  equation  is  subject  to  a  qualification 
connected  with  the  assignment  of  values  to  a. 
It  will  be  recalled  that  that  assignment  was 
somewhat  arbitrary,  and  also  that  the  values 
of  a  entered  into  the  computation  of  the  values 
of  &,  and  7i.  A  systematic  error  in  the  values 
of  a  would  therefore  cause  systematic  errors 
in  the  other  parameters  and  might  vitiate 
conclusions  as  to  the  laws  of  their  variation. 
A  search  was  made  for  evidence  of  such  errors, 
the  search  making  use  of  the  principle  (easily 
demonstrated)  that  a  positive  error  in  a  would 
cause  a  positive  error  in  Jt  and  a  negative  error 
in  7i.  While  the  result  of  the  search  was  nega- 
tive, it  is  not  to  be  supposed  that  the  values  of 
a  have  high  precision.  To  their  errors,  in 
combination  with  the  obscure  influences  of  the 
varying  range  of  fineness  and  with  the  errors  of 
observation,  are  to  be  ascribed  the  irregulari- 
ties of  the  constants  of  the  adjusting  equations. 

While  the  algebraic  relations  are  such  that 
minor  errors  in  the  values  of  a  might  have  im- 
portant influence  on  values  of  bt  and  n,  their 
influence  on  the  interpolated  values  of  C  would 
be  small. 

The  uncertainties  affecting  the  several  ele- 
ments of  equation  (27)  are  so  great  that  no 
attempt  will  be  made  to  develop  from  it  a 
definite  and  quantitative  expression  for  the 
relation  of  capacity  to  slope.  For  this  reason 
the  preceding  paragraphs  have  attempted  to 
present  only  the  general  tenor  of  the  discussion 
20021  °—  No.  SO—  14  -  7 


y  oc  xn 

this  is  the  exact  equivalent  of  the  familar 
"y  varies  as  the  Tith  power  of  x."  This  mode  of 
comparing  the  rate  of  variation  of  one  thing 
with  the  rate  of  variation  of  another  is  exten- 
sively employed,  and  it  so  commends  itself  by 
its  simplicity  that  its  use  is  constantly  extended 
into  fields  where  its  applicability  is  approxi- 
mate only.  Having  occasion  to  make  much  use 
of  certain  variants  of  this  function,  I  find  it 
important  to  obtain  a  clear  conception  of  its 
properties  and  shall  therefore  give  the  matter 
somewhat  elementary  attention  —  with  due 
apology  to  the  mathematical  reader. 

If  we  consider  x  and  y  merely  as  numbers, 
the  rate  of  variation  with  y  with  respect  to  x 
is  the  ratio  of  the  differential  increment  of  y 
to  that  of  x.  That  ratio  is 


.(29) 


If  we  consider  x  and  y  (and  also  the  constant 
a)  as  powers  of  a  common  base,  equation  (28) 
becomes 

log  y  =  log  a  +  n  log  x (30) 

The  rate  of  variation  of  log  y  with  respect  to 
log  x  is,  differentiating, 

-(31) 


dlogx 


98 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


Now,  in  saying  that  one  quantity  varies  as 
a  certain  power  of  another,  or  in  using  such  a 
function  as  (28),  the  index  of  variation,  or  that 
by  which  is  indicated  the  comparative  rate  of 
variation,  is  the  exponent  n;  and  the  value  of 
this  exponent,  in  terms  of  the  variables,  is  found 
not  in  (29)  but  in  (31).  The  expression  "y 
varies  as  the  nth  power  of  x"  is  equivalent  to 
"the  rate  of  variation  of  y,  considered  as  a 
power,  is  n  times  the  rate  of  variation  of  x, 
considered  as  a  power." 


FIGURE  29.—  Logarithmic  locus  of  the  power  function. 

An  equivalent  result  would  be  attained  if 
attention  were  given  to  the  quality  of  the 
growth  of  x  and  y.  Considering  their  growth 
as  occurring  by  additive  increments,  equation 
(29)  gives  the  ratio  between  their  rates  of  in- 
crease. Considering  their  growth  as  a  matter 
of  multiplication  by  ratios,  the  additive  incre- 
ments are  increments  to  logarithms,  and  equa- 
tion (31)  gives  the  ratio  between  rates  of  in- 
crease. 

Equation  (30),  the  logarithmic  equivalent  of 
(28),  is  the  equation  of  a  straight  line.  Repre- 
senting it  by  AB  in  figure  29,  its  inclination 

CO 

equals  n,  and  its  intersection  with  the  axis 


of  log  y  gives  CO  =  log  a.  These  familiar  prop- 
erties enhance  the  utility  of  function  (28)  by 
enabling  the  investigator  to  discuss  its  con- 
stants on  logarithmic  section  paper. 

In  many,  probably  a  large  majority,  of  the 
physical  problems  to  which  the  power  function 
is  applied,  it  is  found  that  the  exponent,  n, 
does  not  have  a  constant  value  through  the 
entire  range  of  observed  values  of  x  and  y. 
The  locus  of  log  y=/(log  x}  is  then  not  a 
straight  line  but  a  curve,  which  we  may  repre- 
sent by  AB  in  figure  30.  At  any  point  of  the 
curve  C,  its  minute  element,  not  distinguishable 

CE 
from  a  straight  line,  has  an  inclination,  jyp 


and  which  we  may  call  n2.  The  value  of  n2 
varies  from  point  to  point  of  the  curve,  so  that 
if  we  try  to  express  the  relation  of  y  to  x  in  the 
form  of  equation  (28)  we  must  regard  the  ex- 
ponent as  a  variable. 

The  element  at  C  may  also  be  thought  of  as 
part  of  the  tangent  h'ne  CD,  of  which  the  equa- 
tion is 

log  y  =  FO  +  ftj 
whence 


On  comparing  this  with  (28),  it  is  seen  that 
log-'FO  corresponds  to  a.  Let  us  replace  it 
by  «,,  giving 

yc,*"-..         (32) 

It  is  evident  that  for  any  other  point  of  the 
curve,  as  C',  the  tangent  intersects  the  axis  of 
log  y  at  a  point  different  from  F,  and  this  cor- 
responds to  a  different  value  of  at.  In  other 
words,  if  we  would  express  in  an  equation  of 
the  type  of  (28)  the  same  relation  between  two 
variables  that  is  expressed  by  the  logarithmic 
locus  in  figure  30,  we  must  make  the  coefficient 
as  well  as  the  exponent  variable. 


which  is  homologous  with  n  in  (31)  and  (28) 


FIGURE  30.— Locus  of  log  y-/(log  x),  illustrating  the  nature  of  the  index 
of  relative  variation. 

The  values  of  a,  and  n2  are  evidently  func- 
tions of  the  independent  variable,  x. 

It  is  possible  to  give  to  the  relation  shown 
by  the  logarithmic  locus  an  algebraic  expression 
which  is  identical  in  form  with  (28)  and  in 
which  the  coefficient  is  a  constant.  That  is  to 
say,  it  is  possible  to  segregate  the  variability  of 
parameters  in  the  exponent;  but  when  that  is 
done  the  exponent  no  longer  corresponds  to  the 
expression  "varies  as  the  nth  power  of,"  and 


RELATION  OF  CAPACITY  TO  SLOPE. 


99 


the  expression  thus  lacks  the  essential  quality 
of  (28).  To  illustrate,  let  us  assume  that  the 
position  of  the  point  F  is  fixed,  so  as  to  give  a 
constant  value  to  a^  The  values  of  n}  then 
correspond  to  the  inclinations  of  lines  FC,  FC' , 
etc.,  drawn  to  points  on  the  curve;  but  these 
lines  are  not  tangents  and  their  inclinations  are 
not  those  of  the  corresponding  elements  of  the 
curve. 

In  order  to  satisfy  the  condition,  from  equa- 
tion (28),  that  y  =  a  when  x=l  and  logz  =  0, 
the  fixed  point  must  be  at  the  intersection,  G, 
of  the  curve  with  the  axis  of  log  y.  The  lines 
connecting  it  with  other  points  of  the  curve  are 
therefore  chords. 

In  the  discussions  of  our  laboratory  data  use 
will  be  made  of  both  these  variants  of  the  power 
function;  and  they  will  be  distinguished,  from 
one  another  as  well  as  from  (28),  by  the  follow- 
ing notation.  In 


y=vxi 


(33) 


the  coefficient  and  exponent  are  both  variable 
and  are  functions  of  x.  The  symbol  v  is  chosen 
for  the  coefficient  to  signalize  its  variability. 
The  exponent,  i,  denotes  the  instantaneous  ratio 
of  the  variation  of  y  to  the  variation  of  x,  when 
those  variations  are  viewed  as  ratios.  It  is  tho 
first  differential  coefficient  of  log  y  with  respect 
to  log  x,  and  it  will  be  spoken  of  as  the  index  of 
relative  variation. 
In 

y  =  cxJ (34) 

the  coefficient  is  constant  and  the  exponent 
variable. 

Whenever,  in  the  investigation  of  the  natural 
law  connecting  two  variables,  pairs  of  simulta- 
neous values  of  the  variables  are  known  by  ob- 
servation, it  is  possible  to  plot  a  curve  represent- 
ing empirically  log  y  =/  (log  2) — such  a  curve  as 
A  CGB  in  figure  30.  The  directions  of  the  ele- 
ments of  that  curve,  or  the  values  of  i,  are  es- 
sentially facts  of  observation.  They  depend 
exclusively  on  the  phenomena  and  are  inde- 
pendent of  the  units  in  which  observational 
data  are  expressed.  It  is  different  with  the 
values  of  j,  for  those  depend  on  the  position  of 
the  point,  G,  in  which  the  curve  intersects  the 
axis  of  log  y,  and  therefore  on  the  position  of 
the  axis.  The  position  of  the  axis  corresponds 


to  log  x  =  0  or  x  =  1  and  is  thus  dependent  on 
the  magnitude  of  the  unit  by  which  the  inde- 
pendent variable  is  measured. 

In  other  chapters  of  this  report  much  atten- 
tion is  given  to  the  values  of  i,  and  the  discus- 
sion of  the  variations  of  such  values  is  used  as 
a  mode  of  treating  empirically  the  relations 
between  the  various  factors  of  the  general 
problem  of  traction. 

THE    SYNTHETIC    INDEX. 

Recurring  to  figure  30,  let  us  give  attention  to 
a  restricted  portion  of  the  curve,  for  example, 
the  part  between  C  and  C1 .  The  value  of  i  cor- 
responding to  the  point  C  is  the  inclination  of 
the  line  OF;  the  value  of  i  corresponding  to  C' 
is  the  inclination  of  C' F'.  Between  the  two  are 
a  continuous  series  of  other  values.  The  incli- 
nation of  the  chord  connecting  C  and  C',  con- 
considered  as  a  ratio  or  exponent,  is  interme- 
diate between  the  extreme  values  of  i.  If  the 
sequence  of  values  follows  a  definite  law,  the 
value  given  by  the  chord  equals  some  sort  of  a 
mean  derived  from  the  others ;  and,  in  any  case, 
it  is  in  a  sense  representative  of  the  group.  It 
may  be  called  a  synthetic  index  of  relative  varia- 
tion between  the  indicated  limits. 

If  the  coordinates  of  C  be  log  x'  and  log  y', 
and  the  coordinates  of  C'  be  log  x"  and  log  y", 
then,  representing  the  synthetic  index  by  7, 


.(35) 


/= logy" -log  y' 
log  x"— log  x' 

As  the  direction  of  the  chord  depends  on  the 
positions  of  G  and  G'  upon  the  curve,  so  the 
value  of  7  depends  on  the  limits  between  which 
it  is  computed.  As  the  direction  of  the  chord 
gives  no  information  concerning  the  direction 
of  any  part  of  the  curve,  so  the  value  of  7  can 
not  be  used  to  determine  any  particular  value 
of  i.  It  is  used  in  the  following  pages  for  the 
comparison  of  different  functions  for  which  the 
data  span  approximately  the  same  range  of 
conditions. 

APPLICATION   TO    THE    SIGMA    FUNCTION. 

Let  us  now  represent  the  relation  of  capacity 
to  slope  by  an  equation  of  form  (33), 

G=VlS''..  ..(36) 


100 


TRANSPORTATION    OF   DEBRIS   BY    RUNNING    WATER. 


and  develop  the  value  of  it  from  equation  (10). 
The  logarithmic  equivalent  of  (10)  (p.  64)  is 

log  C"=log  6,+nlog  (S-a) (15) 

Differentiating,  and  dividing  both  -members  by 
d  log  8, 


Introducing  these  values  in  (37)  and  reducing, 
we  have 


tl~S-<j' 


(38) 


.(37) 


d  log  G= n d\og(S-a) 
d  log  S      d  log  S 


The  first  member  of  this  equation  is  the  value 
of  il}  and  it  may  be  substituted  for  it.     Also 


and 


The  values  of  n  and  a  being  known  for  each 
observational  series,  it  is  possible,  by  means  of 
equation  (38),  to  compute  \  for  any  value  of  S. 
The  values  of  il  have  been  computed  for  series 
of  values  of  S  having  the  uniform  interval  of 
0.2  per  cent  (and  below  1.0  per  cent  the  inter- 
val of  0.1  per  cent)  and  having  such  range  in 
each  observational  series  as  to  correspond  with 
the  range  of  the  observations.  These  values 
are  recorded  in  Table  15.  It  is  proposed  to 

,,       o  _dS  discuss  their  relations  to  various  conditions, 

S  beginning  with  slope. 

TABLE  15. —  Value*  ofilt  the  index  of  relative  variation  for  capacity  in  relation  to  slope. 


dlog  (S-a)= 


Conditions  of  experimentation  

Grade.  . 

(A) 

w  

0.66 

1.00 

1.32 

1.9fi 

Q-.   . 

0.093 

0.182 

0.545 

0.182 

0.363 

0.734 

0.182 

0.363 

0.734 

0.363 

0.734 

1.119 

Parameters  of  adjusting  equation  .  . 

&;::::: 

n 

0.40 
32.2 
1.50 

0.20 
66.2 
1.60 

0.04 
150.3 
1.42 

0.12 
47.4 
1.81 

0.07 
113.3 
1.70 

0.05 
252.3 
1.70 

0.17 
50.1 
1.80 

0.10 
122.8 
1.53 

0.07 
282.4 
1.59 

0.17 
128.9 
1.58 

0.10 
280.1 
1.40 

0.08 
397.0 
1.21 

S 

Values  of  u. 

0.2  

3.05 

2.79 

.4 

2.07 
1.98 

1.92 
1.89 

.87 
.85 
.83 

.81 
.79 

:78 
.77 

.94 
.89 

.85 
.83 
.81 
1.80 
1.79 

1.78 

2.29 
2.04 
1.91 

1.83 

1.78 
1.75 
1.72 
1.70 

1.67 
1.65 
1.63 
1.62 
1  61 

2.08 
1.93 
1.85 

1.80 
1.77 
.74 
.72 
.71 

.69 
.68 

"'2.'  74' 
2.39 

2.20 
2.08 
2.00 
1.94 
1.90 

.84 

.80 
.76 
.74 
72 

2.09 
1.86 
1.74 

1.68 
1.63 
1.60 
1.57 
1.55 

1.52 

.5  

1.54 

.52 
.50 
.49 
.48 
.48 

1.47 

2.39 

2.27 
2.19 
2.13 
2.09 
2.06 

2.02 
1.99 
1.96 
1.95 
1.93 

2.72 

2.51 
2.38 
2.28 
2.22 
2.17 

2.10 
2.05 
1.02 
1.99 
1  97 

1.63 

1.57 
1.53 
1.50 
1.48 
1.47 

1.44 

1.43 
1.42 
1.41 
1.40 

1.39 

g 

.7...                           

2.24 
2.12 
2.05 
1.99 

1.92 

1.86 
1.82 
1.79 

.8... 

.9  

1.0  

a.  2... 

2.25 
2.10 
2.00 
1.93 

1.88 

1.4  

1.6  



1.8  

2.0  

2.2  

1.84 

1.92 

1  95 

BELATION    OF   CAPACITY   TO   SLOPE. 
TABLE  15. —  Values  o/t,,  the  index  of  relative  variation  for  capacity  in  relation  to  slope — Continued. 


101 


Crade.. 

(ID 

Condi!  ions  of  experimentation  to  

0.23 

0.44 

0.68 

1.00 

o 

0.093 

0.182 

0.093 

0.182 

0.093 

0.182 

0.363 

0.545 

0.182 

0.393 

0.545 

0.06 
157.2 
1.53 

0.734 

|»  
Parameters  of  adjusting  equation  .  .  tt>\  
(n 

0.70 
23.8 
0.93 

0.60 
33.2 
0.99 

0.50 
29.7 
1.15 

0.10 
34  3 
1.63 

0.30 
19.4 
1.64 

0.10 
39.8 
1.64 

0.08 
91.4 
1.45 

0.07 
132.8 
1.38 

0.12 
37.6 
1.69 

0.08 
97.5 
1.54 

0.06 
219.3 
1.52 

S 

Values  of  d. 

0.  2                       

2.13 

1  80 

2.57 
2  10 

2.17 

1.89 
1.78 
1.72 

.69 
.66 
.64 
.62 
.61 

.60 
.58 
.58 
.57 

3                                     

2.46 
2.19 
2.05 

.97 
.92 
.88 
.85 
.82 

.79 
.77 
.75 
.74 
.73 

.72 

4                         

1.68 

1  93 

5                                   

2.08 

1.73 

.67 
.64 
.61 
.59 
.58 

.55 
.54 
.53 
.52 
.51 

1.61 

1.57 
1.54 
1.51 
1.50 
1.49 

1.47 
1.46 
1.45 
1.44 
1.43 

2.22 

2.11 
2.04 
1.98 
1.95 
1.92 

1.88 
1.85 
1.83 
1.81 
1.80 

1  79 

1.84 

1.78 
1.74 
1.71 
1.69 
1.68 

1.65 
1.63 
1.62 
1.61 
1.60 

1  60 

6                                

1.96 

7                                            

6.96 
3.98 
2.98 
2.49 

1.99 
1.74 
1.59 
1.49 

4.04 
3.08 
2.60 
2.31 

1.98 
1.80 
1.68 
1.60 
1.54 

1.90 
1.86 
1.83 
1.81 

1.78 
4.76 
1.74 
1.73 

2.87 
2.63 
2.46 
2.34 

2.19 
2.09 
2.02 
.97 
.93 

.90 

.8                              

7.41 
4.17 
3.10 

2.22 
1.85 
1.65 
1.52 
1.43 

.65 
.63 
.62 

.61 
.59 
.58 
1.58 
1.57 

1  57 

9                                                

1  o                          

1.2                  

14                                        

1.6                  

1  8                                  

2  0                                                 

2  •>                                                 

1.36 

1.50 

2.  4                                      

1.46 

.88 

.71 

1.78 
1  77 

1.59 

2  6                                         

1.43 

.86 

.71 

2.  8                                

1.84 

.70 

1.76 

3  0                                          

1.82 

.70 

1.76 

3.2                                                  

.69 

3.4  

.69 

3  6                                                1  

.69 

3  8                                                                     

.69 

4.  0                                          

.68 

4.  2                                      

1.68 

Grade.  . 

(B) 

(C) 

1.32 

1.96 

0.44 

0.66 

Q 

0.182 

0.363 

0.545 

0.734 

0.363 

0.545 

0.734 

1.119 

0.093 

0.182 

0.093 

0.182 

0.17 
39.5 
1.61 

0.12 
96.5 
1.54 

0.10 
165.4 
1.55 

0.08 
233.5 
1.57 

0.18 
93.9 
1.64 

0.14 
153.5 
1.60 

0.12 
228.2 
1.46 

0.10 
363.3 
1.43 

0.40 
22.6 
1.43 

0.20 
34.4 
1.50 

0.16 
18.2 
1.58 

0.11 
38.6 
1.54 

Parameters  of  adjusting  equation  .  .  fti  

S 

Values  of  ii. 

0  2 

2.86 

3 

2.56 

2.14 
1.96 

2.44 

2.15 

2.43 
2.12 
1.97 

1.88 
1.83 
1.79 
1.75 
1.73 

1.70 

1.67 
1.65 
1.64 
1.63 

1.62 
1.61 
1.61 
1.60 
1.60 

1.59 
1.59 
1.59 
1.59 

4 

2.80 
2.44 

2.25 
2.13 
2.05 
1.99 
1.94 

.88 

.84 
.80 
.78 

2.20 
2.02 

1.92 

1.85 
1.81 
1.77 
1.75 

1.71 
1.68 
1.66 
1.64 

2.99 

2.09 
1.93 

1.83 
1.77 
1.72 
1.69 
1.66 

1.63 
1.60 
1.58 
1.57 

1.91 
1.79 

1.73 
1.68 
1.64 
1.62 
1.60 

1.57 

g 

1.94 

1.86 
1.81 
1.78 
1.75 
1.73 

1.70 
1.67 
1.66 
1.65 

1.87 

1.81 
1.78 
1.75 
1.72 
1.71 

1.68 
1.67 
1.65 
'    1.64 

2.57 

2.35 
2.21 
2.12 
2.06 
2.01 

.94 

.89 
.85 
.83 

2.23 

2.09 
2.01 
.94 
.90 
.86 

.82 
.78 
.76 
.74 

2.33 

2.16 
2.05 
1.98 
1.92 

1.87 

1.82 
1.79 
1.76 
1.74 
1.72 

1.70 
1.69 
1.69 
1.68 
1.67 

1.66 
1.66 
1.65 
1.65 
1.65 

6                                                          

4.29 
3.39 
2.86 
2.57 
2.39 

2.15 
2.00 
1.91 
1.84 

2.22 
2.10 
2.00 
1.93 
1.88 

1.81 
1.76 
1.72 
1.69 
1.67 

1.65 

8                                                   

9 

1  o                                       

1  2 

1  4 

1.6  
1.8  

2.0  

2  2 

.76 
.75 

1.63 

1.62 

.81 
1.79 

1.56 

1.79 
1.75 

2  4 

74 

2  6 

2  8 

3  0 

| 

32 

3  4 

3  6 

3  8 

4  0 

4  2 

1.64 

102  TRANSPORTATION   OF   DEBRIS   BY   RUNNING    WATER. 

TABLE  Jo. —  Values  of  ilt  the  index  of  relative  variation  for  capacity  in  relation  to  slope — Continued. 


Grade.  . 

(C) 

Conditions  of  experimentation  w  

0.66 

1.00 

1.32 

Q 

0.363 

0.545 

0.08 
122.8 
1.48 

0.734 

0.182 

0.363 

0.545 

0.734 

1.119 

0.182 

0.363 

0.545 

0.734 

{t 

0.08 
82.9 
1.46 

0.04 
162.1 
1.50 

0.15 

40.4 
1.59 

0.11 
100.1 
1.41 

0.09 
159.2 
1.33 

0.07 
199.2 
1.35 

0.06 
301.6 
1.39 

0.22 
33.7 
1.85 

0.  16         0.  13 
99.9       156.1 
1.40         1.35 

0.11 
218.5 
1.30 

bi  
n... 

S 

Values  of  fi. 

0  2 

2.12 

86 

3.14 

2.18 
1  76 

2.90 
2.12 
1.80 
1.67 

1.60 
1.55 
1.51 
1.49 
1.47 

1.44 
1.42 
1.40 
1.39 
1.38 

1.37 

3 

2.23 

3.00 

4 

1.83 
1.74 

1.68 
1.65 
1.62 
1.60 
1.59 

1.56 

.75 
.69 

.65 
.62 
.60 
.59 
.58 

.56 

2.54 
2.27 

2.12 
2.02 
.95 
.90 
.87 

.81 
.78 
.75 
73 

1.95 
1.81 

1.73 
.68 
.64 
.61 
.59 

.55 

.53 
.51 
50 

1.64    

2.33 

5 

1.63 

.61 
.59 
.58 
.57 
.57 

1.63 

1.57 
.53 
.50 
.48 
.47 

.44 
.43 
.42 
41 

1.57 

1.53 
1.50 
1.48 
.46 
.45 

.48 
.42 
.41 
.41 

1.58 

1.55 

.52 
.50 
.49 
.48 

.46 

.45 

2.06 

.6.... 

2.92 
2.70 
2.55 
2.45 
2.37 

2.27 
2.20 
2.14 
2.11 
2.08 

2.06 
2.04 

1.91 
1.82 
.75 
.70 
.67 

.61 
.58 
.55 
.54 
.52 

.51 
.50 

1.72 
1.65 
1.61 
1.57 

1.55 

1.51 
1.49 
1.47 
1.45 

1.44 

1.43 
1.42 

7    . 

g 

9  .     . 

1  0 

1.2.... 

14..                                         ... 

1.55 
1.54 
1.53 
1.52 

1.52 
1  51 

.55 
.54 
.54 
.53 

1.53 

1  6 

1  8 

2  0 



.71 

.70 
69 

.49 

.49 

48 

.40 
1.39 

1.40 
1.39 



2  2 

2  4 

2  6 

1.51 

.68 

.47 

28  

.68 

.47 

3  0 

3  2. 

3  4 

36...                          

3  8 



Grade.  . 

(C) 

(D) 

1.96 

0.66 

0.182 

1. 
0.363 

00 

Q  

0.363 

0.545 

0.734 

1.119 

0.093 

0.182 

0.545 

0.545 

0.734 

Parameters  of  adjusting  equation  

S:::::: 

0.24 
93.4 
1.62 

0.20 
155.7 
1.50 

0.17 
245.1 
1.34 

0.14 

438.3 
1.59 

0.19 
13.2 
1.80 

0.14 
37.2 
1.55 

0.08         0.17 
115.2         32.8 
1.51          1.67 

0.12 
87.9 
1.49 

0.10 
129.4 
1.68 

0.08 
174.6 
1.52 

S 

Values  of  ii. 

0.2                           

2.52 

2.54 

2.12 
1.90 
1.81 

1.75 
1.72 
1.6» 
1.67 
1.65 

1.63 
1.61 
1.60 
1.59 

3 

<i  m 

2  48 

.4                                     

2.33 

2.38 
2.15 

2.02 
1.93 
1.88 
1.83 
1.80 

.75 
.72 
.70 
.68 
.66 

.65 
.64 
.63 

1.89 
1.80 

1.74 
1.71 
1.68 
1.66 
1.64 

1.62 

1.60 
1.59 
1.58 
1.57 

1.57 

2.12 
1.95 

1.86 
1.80 
1.75 
1.71 
1.69 

1.65 
1.63 
1.61 
1.59 
1.58 

1.57 
1  56 

"'i.'ii' 

2.04 
1.97 
1.93 
1.90 
1.87 

1.84 
1.81 
1.80 
1.79 
1.78 

.5  

3.12 

2.71 
2.47 
2.32 
2.21 
2.14 

2.03 
1  96 

2.49 

2.24 
2.09 
.99 
.92 
.87 

.79 
74 

2.03 

.87 
.77 
.70 
.65 
.62 

.56 
53 

2.21 

2.07 
1.99 
1.93 
1.88 
1.85 

1.80 

.6 

2.64 
2.47 
2.36 
2.28 
2.22 

2.14 

2.08 
2.04 
2.01 
1.99 

1.97 
1.96 
1.94 

2.20 
2.12 
2.06 
2.01 

1.94 
1.90 
1.86 
1.84 
1.82 

1.81 

1  80 

7 

.8       

9 

1.0           

1.2... 

1  4 

16                                  ... 

1.91 
1  87 

.71 
68 

.50 

1  8 

20                             .              .... 

1  84 

.66 

2.2 

1  82 

1  65 

2.4  

2  6 

1  79 

J.8  i  

1  78 

RELATION   OF   CAPACITY   TO   SLOPE. 
TABLE  15. —  Values  of  /,,  the  index  of  relative  variation  for  capacity  in  relation  to  slope — Continued. 


103 


Grade. 

(D) 

(E) 

1.32 

0.6«                                          1.00 

1.32 

Q 

0.363 

0.734 

0.363 

0.734 

0.182 

0.363 

0.734 

1.119 

0.363 

0.734 

1.119 

{"... 

0.16 
SO.  1 

1.77 

0.11 

160.0 
1.65 

0.16 
33.8 
1.93 

0.11 
50.2 
1.73 

0.08 
17.0 
1.72 

0.06 
37.3 
1.70 

0.04 
76.5 
1.58 

0.03 
145.9 
1.58 

0.04 
38.8 
1.57 

0.03 
76.1 
1.50 

0.02 
126.1 
1.77 

ti 

n. 

s 

Values  of  /i. 

0.2... 

2.43         1.98 

2  13         i  M 

.3  

2.60 

.4  

2.27 

2  16 

2  00 

1  76 

.5  

2.60 

2.41 
2.29 
2.21 
2.15 
2.11 

2.04 
2.00 

2.11 

2.02 
1.95 
1.91 
1.88 
1.85 

1.81 

2.06 

.99 
.95 
.92 
.90 
.88 

.85 
.83 
.82 
.81 
1.80 

1.93 

.89 
.86 
.84 
.82 
.81 

.79 

.78 
.77 
.76 
.76 

1.72 

1.69 
1.68 
1.67 
1.66 
1.65 

1.64 
1.63 
1.62 
1.62 
1.61 

1.69 

1.67 
1.65 
1.64 
1.64 
1.63 

1.62 
1.62 
1.61 
1.61 
1  61 

1  59 

.6 

1.68 
1.67 
1.66 
1.65 
1.64 

.63 
.62 
.61 
.61 
60 

1.S7 
1.56 
1.55 
1.55 
1.54 

1.54 

1.83 
1.82 
1.82 
1.81 
1.81 

1.80 

.8 

.9... 

1.0 

2.09 

2.03 
1.99 

2.17 

2.13 

2.10 
2.08 

1.2 

1.4  

1.6  

1  97 

1.8  

1.94 

2.  0  

1.92 

2.2... 

1.79 

1.75 

1.61 

2.4  

1  79 

1  75 

1  60 

2.6  

1.78 

2.8  

1  78 

Grade.. 

(F) 

(G) 

0.66 

1.00 

1.32 

0.66 

Q 

0.182 

0.363 

0.734 

0.182 

0.363 

0.734 

1.119 

0.363 

0.734 

1.119 

0.363 

Parameters  of  adjusting  equation  H 

a  
61  

0.33 
12.02 
1.86 

0.21 
30.5 
1.69 

0.17 
52.2 
1.61 

0.44 

7.96 
2.31 

0.31 
28.2 
1.88 

0.22 
65.0 
1.61 

0.17 
106.9 
1.60 

0.39 
30.3 
1.63 

0.28 
71.1 
1.73 

0.23 
126.4 
1.49 

0.50 
27.5 
1.74 

S 

Values  of  /  . 

0.2... 

.3  

.4  

.5  

i 

.6... 

i' 

.7  

2.35        2.08 

.8  

3  07 

2  22        2  fin 

2.65 
2.50 
2.40 

2.25 
2.15 
2.09 

2.09 
2.00 
1.94 

1.85 
1.78 
1.74 
1.71 

.9  

98 

2.87 
2.72 

2.54 
2.42 
2.33 
2.27 
2.23 

2.19 

2.13 
2.06 

1.98 
1.91 
1.87 
1.84 

.94 
.90 

.83 
.79 
.76 
.74 

1.0  

2.14 

2.04 
1.99 
1.94 
1.91 
1.89 

.94 

.88 
.83 
.80 
.78 

3.65 
3.38 
3.20 
3.07 
2.97 

2.90 

2.42 
2.27 
2.16 
2.08 
2  03 

3.48 

2.98 
2.71 
2.53 
2.41 
2.32 

2.25 
2.20 
2.16 
2.12 
2.08 

1.2... 

2.57 
2.44 
2.35 
2.28 
2.23 

2  19 

1.4  

1,6  

1.8  

2.0  

2.2... 

1  99 

2.4  

2.18 

1.84 

2.16 

2.6  

2.14 

1.79        2.14 

2.8  

3.0  

.j 

104  TRANSPORTATION    OF    DEBRIS   BY    RUNNING    WATER. 

TABLE  15. —  Values  of  ilt  the  index  of  relative  variation  for  capacity  in  relation  to  slope — Continued. 


Grade. 

(G) 

(H) 

0.66 

1.00 

1.32 

0.6t> 

d  

0.734 

1.119 

0.363 

0.734 

1.119 

0.363 

0.734 

1.119 

0.363 

0.734 

1.119 

XT  

0.36 
64.5 
1.69 

0.28 
85.0 
1.78 

0.58 
23.6 
1.85 

0.41 

65.7 
1.70 

0.33 
115.5 
1.63 

0.71 
15.39 
2.37 

0.50 
62.3 
1.71 

0.40 
114.5 
1.54 

0.80 
19.38 
2.01 

0.56 
52.9 
1.72 

0.48 
75.1 
1.67 

n  

s 

Values  of  u. 

0.2... 

| 

1 

.3 

i 

4 

.5 

g 

3  23 

3.63 

.7.. 

3.49 
3  08 

2.89 
2  68 

4.11 
3.49 
3.12 
2.88 

2.58 
2.40 
2.29 
2.20 
2.14 

2.09 
2.05 
2.02 

3.09 
2.78 
2.58 
2.44 

2.25 
2.14 
2.06 
2.00 
1.96 

1.92 
1.89 

3.61 

5.31 

4.17 
3.58 
3.21 

2.78 
2.54 
2.38 
2.27 
2.20 

2.14 
2.09 
2.05 

8 

4.58 
3.86 
3.43 

2.94 
2.64 
2.49 
2.38 
2.29 

2.22 
2.17 
2.12 
2.08 

3.09 
2.78 
2.57 

2.32 
2.16 
2.06 
.99 
.93 

:S 

.83 

5.75 
4.56 
3.92 

3.23 

2.88 
2.65 
2.50 
2.40 

2.32 
2.25 
2.20 
2.16 
2.12 

.9... 

2.82 
2.65 

2.42 
2.28 
2.19 
2.12 
2.07 

2.03 
1.99 
1.96 

2.54 
2.43 

2.29 
2.20 
2.14 
2.09 
2.05 

2.02 
2.00 
1.98 

1  0 

1  2 

3.57 
3.15 
2.90 
2.71 
2.60 

2.51 
2.43 
2.38 
2  33 

6.03 

4.69 
4.02 
3.62 
3.35 

3.16 

3.01 
2.90 
2.81 

1.4 

1.6  

1.8 

3.91 
3.67 

3.50 
3.36 
3.26 
3.18 
3.11 

2.0  .... 

2.2.. 

2  4 

2.6.   .. 

2  8 

3.0... 

2.29 

3.2.. 

3.05 

2  09 

3  4 

3  00 

VARIATION    OF    THE    INDEX. 

Each  column  of  the  table  contains  a  set  of 
values  of  tt  which  pertain  to  the  same  grade, 
fineness,  width,  and  discharge  and  of  which  the 
changes  are  related  to  slope  only.  In  figure  31 
a  number  of  these  sets  are  plotted  in  relation  to 
slope.  The  curves  have  a  strong  family  like- 
ness, arising  from  the  fact  that  the  data  were 
all  adjusted  by  the  sigma  formula;  but  the 
likeness  would  not  altogether  disappear  if  the 
assumptions  of  that  formula  were  abandoned. 
The  general  relations  of -the  index  to  slope  are 
as  follows: 

(1)  It  varies  decreasingly  with  slope. 

(2)  Its  rate   of  change   is   greater   for  low 
slopes  than  for  high. 

The  upper  group  of  curves  all  pertain  to 
grade  (C)  and  width  1.00  foot,  but  represent 
different  discharges.  They  show  (3)  that  the 
rate  of  change  for  similar  slopes  is  greater  for 
small  discharges  than  for  large. 

The  second  group  of  curves  all  pertain  to 
grade  (C)  and  discharge  0.363  ft.3/sec.,  but 
represent  different  widths.  They  show  (4) 
that  the  rate  of  change  for  similar  slopes  is 
greater  for  broad  channels  than  for  narrow,  or, 


as  the  depth  varies  inversely  with  the  width, 
that  the  rate  of  change  is  greater  for  shallow 
streams  than  for  deep. 

The  third  group  of  curves  all  pertain  to 
width  1.00  foot  and  discharge  0.363  ft.3/sec., 
but  represent  different  grades  of  debris.  They 
show  (5)  that  the  rate  of  change  is  greater  for 
coarse  debris  than  for  fine. 

In  the  third  group  the  curves  for  grades  (A), 
(B),  (C),  (D),  and  (E)  lie  close  together,  while 
those  for  the  coarser  grades  (F)  and  (G)  are 
well  separated.  This  is  probably  connected 
with  the  fact  that  the  range  of  fineness  grad- 
ually increases  from  (A)  to  (E)  and  then  drops 
abruptly  from  (E)  to  (F).  The  influence  of 
increasing  range  approximately  neutralizes 
that  of  decreasing  fineness,  and  the  inference 
is  (6)  that  the  rate  of  change  in  the  index  is 
greater  for  small  range  than  for  large. 

Consider  now  the  variations  of  the  index  in 
relation  to  width.  In  figure  32  (p.  106)  the 
ordinates,  as  before,  represent  values  of  \  and 
the  abscissas  represent  width  of  channel.  The 
points  fixed  by  the  data  are  shown  by  the  dots. 

(7)  The  upper  group  of  curves  all  pertain  to 
grade  (C)  and  discharge  0.182  ft.3/sec.,  but  rep- 
resent different  slopes.  Their  common  char- 


RELATION  OF  CAPACITY  TO  SLOPE. 


105 


actor  is  a  distinct  minimum.  From  the  neigh- 
borhood of  width  0.66  foot  there  is  increase  of 
it  in  the  direction  of  greater  width,  and  also  in 
the  direction  of  less  width. 

(8)  The  position  of  the  minimum  is  appar- 
ently the  same  for  low  slopes  as  for  high. 


(9)  The  minimum  is  most  strongly  marked 
in  case  of  the  gentler  slopes. 

The  second  group  of  curves  all  pertain  to 
grade  (C)  and  a  slope  of  1.0  per  cent  but  differ 
in  respect  to  discharge.  Each  of  them  shows 
a  minimum,  except  the  curve  for  discharge 


Grade 


Grade  (CJ 


'O  W-lft. 


W-lft 

0. 


363  ft.^ec. 


I  Z 

Slope 

FIGURE  31.— Variations  of  i'i  in  relation  to  slope. 


0.093  ft.3/sec.,  which  has  but  two  fixed  points. 
They  show  also  that— 

(10)  The  position  of  the  minimum  is  related 
to  discharge.     For  large  discharges  it  is  asso- 
ciated with  relatively  large  widths,  for  smaller 
discharges  with  smaller  widths. 

(11)  The  minimum  is  more  pronounced,  or 
the  associated  rates  of  change  in  the  index  are 
higher  in  case  of  small  discharge  than  of  large- 


Various  analogies,  which  appear  in  another 
part  of  this  paper,  render  it  probable  that  all 
the  preceding  inferences  are  of  a  general  char- 
acter; but  those  in  regard  to  width  are  not  sus- 
tained by  all  the  data. 

The  curves  of  the  third  group  are  based  on 
observations  with  grade  (B)  and  are  drawn, 
like  those  of  the  first  group,  to  contrast  the 
relations  of  the  index  to  width  for  different 


106 


TRANSPORTATION    OF    DEBRIS   BY   RUNNING    WATER. 


slopes.  With  the  gentler  slopes  they  give 
indications  of  a  minimum,  but  not  with  the 
steeper.  The  character  of  the  discrepancy  is 
such  as  to  suggest  that  the  values  of  the  index 
computed  for  width  0.23  foot  vary  too  rapidly 
with  slope;  and  this  result  might  be  brought 
about  by  assigning  too  large  a  value  to  a.  A 
critical  review  of  the  data,  however,  failed  to 
find  warrant  for  any  material  change  in  that 
constant. 

It  is  believed  that  a  group  of  discrepancies 
which  this  instance  illustrates  are  connected 


with  the  history  of  the  experimental  work. 
The  first  grade  to  be  investigated  was  (B),  and 
the  methods  of  manipulation  were  subject  to 
various  minor  changes,  which  were  not  always 
recorded;  but  the  range  of  conditions  was  large. 
Grade  (C)  was  next  taken  up,  and  again  the 
range  of  conditions  was  large.  Other  grades 
followed,  with  less  elaborate  range  of  condi- 
tions; but  the  work  on  grade  (G)  was  somewhat 
expanded,  in  order  to  learn  the  influence  of 
coarser  debris  on  various  factors.  The  work 
on  (G)  also  had  the  advantage  of  the  fullest 


.66 


1.32 


1.00 
Width 

FIGURE  32.— Variations  of  ii  in  relation  to  width  of  channel. 


1.96 


development  of  experimental  method  as  well  as 
that  of  uniformity  of  method.  Because  of  this 
history  it  is  believed  that  the  results  for  grades 
(C)  and  (G)  are  of  higher  authority  than  those 
of  other  grades;  and  the  belief  is  strengthened 
by  the  general  symmetry  and  internal  con- 
sistency of  the  (C)  and  (G)  results.  The  infer- 
ences, given  in  preceding  paragraphs,  from 
data  of  grades  (C)  and  (G)  are  therefore  ac- 
cepted, and  the  discordance  of  data  for  grade 
(B),  while  not  specifically  explained,  is  ascribed 
in  a  general  way  to  unrecorded  differences  in 
laboratory  methods. 


The  curves  of  the  fourth  group  of  figure  32 
all  pertain  to  grade  (G)  and  discharge  0.734 
ft.3/sec.  but  differ  in  respect  to  slope.  Com- 
pared with  the  first  and  second  groups  they  are 
seen  to  be  consistent  with  the  inference  as  to  a 
minimum  value  of  %,  but  the  minimum  falls 
outside  the  range  of  widths  for  which  data  were 
obtained.  With  grade  (C)  and  discharge  0.734 
ft.3/sec.  the  minimum  falls  between  widths  1 
foot  and  1.32  feet,  but  nearer  to  the  former. 
With  grade  (G)  and  the  same  discharge  it  ap- 
parently falls  with  some  width  less  than  0.66 
foot.  This  indicates  that — 


BELATION  OF  CAPACITY  TO  SLOPE. 


107 


(12)  The  position  of  the  minimum  is  related 
to  fineness.  For  the  finer  debris  it  is  associated 
with  relatively  great  width;  for  the  coarser, 
with  smaller  width. 

The  curves  of  the  fourth  group  support  the 
ninth  inference,  that  the  minimum  is  most 
strongly  marked  for  the  gentlest  slope. 


In  the  study  of  the  data  many  other  com- 
parisons of  the  influence  of  width  were  made, 
but  they  are  not  here  illustrated.  Their  chief 
service  was  in  indicating  the  comparative  value 
of  different  divisions  of  the  body  of  data.  The 
general  fact  brought  out — and  one  emphasized 
in  various  other  ways — is  that  the  measures  of 


.093     .183 


.363 


.734 


.545 
Discharge 

FIGURE  33.— Variations  of  ij  in  relation  to  discharge. 


1. 119 


precision  derived  from  discrepancies  of  observa- 
tions within  a  single  series  by  no  means  cover 
the  whole  field.  The  discrepancies  discovered 
when  properties  of  different  series  are  compared 
are  quite  as  important  and  must  be  given  con- 
sideration in  connection  with  the  broader  gen- 
eralizations. 

Let  us  now  consider  the  relations  of  the  vary- 
ing value  of  ij  to  discharge.     These  are  illus- 


trated by  figure  33.  The  curves  of  the  upper 
group  all  pertain  to  grade  (C)  and  slope  1.0 
per  cent  but  differ  in  respect  to  width  of  chan- 
nel. Those  of  the  second  group  pertain  to 
grade  (C)  and  slope  1.8  per  cent;  those  of  the 
third  group  to  grade  (G)  and  slope  1 .8  per  cent. 
The  general  fact  is  that — 

(13)  As  discharge  increases  the  value  of  it  di- 
minishes.    There  are  three  exceptions,  of  which 


108 


TRANSPORTATION    OF    DEBRIS    BY    RUNNING    WATER. 


two  do  not  exceed  the  computed  probable 
errors  of  the  data,  and  the  third  is  connected 
with  a  value  of  \  to  which  the  lowest  weight 
is  ascribed. 

(14)  The  rate  of   change  in  the  index  is 
greater  for  small  discharges  than  for  large. 

(15)  For  the  same  discharges  the  rate  of 
change  in  the  index  is  greater  for  wide  chan- 
nels than  for  narrow,  and  is  therefore  greater 
for  shallow  streams  than  for  deep. 


FIGURE  34.— Variations  of  ii  in  relation  to  fineness  of  de"bris. 

The  curves  of  the  lowest  group  all  pertain  to 
width  0.66  foot  and  slope  1.8  per  cent,  but 
differ  as  to  grade  of  debris.  They  indicate 
that— 

(16)  The  rate  of  change  in  the  index  is 
greater  for  coarse  debris  than  for  fine.  The 
peculiarities  of  spacing,  as  in  a  previous  in- 
stance, may  show  the  influence  of  the  factor  of 
range  in  fineness  within  the  several  grades. 

To  consider  now  the  relations  of  the  values 
of  \  to  the  fineness  of  debris,  the  comparison  is 
made  with  linear  fineness — instead  of  bulk  fine- 
ness, as  in  discussing  a,  and  it  is  found  con- 
venient to  plot  the  logarithms  of  the  quantities 
instead  of  the  quantities  themselves.  In  figure 


34  the  curves  of  the  upper  group  are  derived 
from  experiments  conducted  with  a  trough 
width  of  0.66  foot,  and  each  one  pertains  to  a 
particular  combination  of  slope  and  discharge. 
Those  of  the  second  group  are  derived  from  ex- 
periments with  a  trough  width  of  1  foot. 

(17)  In  the  main  they  show  decrease  of  it 
with  increase  of  fineness,  but  the  finer  grades 
give  the  opposite  indication.  The  data  are  not 
sufficiently  harmonious  to  determine  whether 
the  law  of  change  is  continuous  or  involves  a 
reversal.  If  it  is  continuous,  i{  is  an  inverse 
function  of  F. 

In  view  of  the  fact  that  the  double  variation 
of  \  in  relation  to  width  is  a  complicating  factor 
and  of  the  further  fact  that  that  variation  is 
less  pronounced  with  high  slopes  than  with  low, 
two  curves  (the  lowest  group  of  fig.  34)  were 
constructed  from  data  pertaining  to  the  highest 
practicable  slope,  2.4  per  cent.  Each  curve 
belongs  to  a  particular  width  of  trough,  and 
each  is  a  composite  with  respect  to  discharge. 
Their  indication  is  practically  the  same  as  that 
of  the  other  groups.1 

The  character  of  the  material  has  not  seemed 
to  warrant  a  quantitative  discussion  of  the 
variations  of  the  index  of  variation,  and  a  sum- 
mary of  the  qualitative  discussion  is  neces- 
sarily limited  to  generalities.  The  index  of 
relative  variation  or  the  sensitiveness  of  capac- 
ity for  traction  to  change  of  slope  is  a  decreas- 
ing function  of  the  slope,  the  discharge,  the 
fineness  of  debris,  and  the  range  of  fineness 
and  is  a  minimum  function  of  width  of  channel. 

In  symbols, 

»!=/$,&  F,&,w)  ---  .....  (39) 

If  we  assume  tentatively  that  the  function  re- 
placing it  in  the  exponent  is  the  product  of 
functions  of  the  individual  conditions  —  that  is, 
if  we  write 


then  we  must  also  recognize  that  in  /,(£),/,  is 
itself  a  function  of  Q,  F,  H,  and  w,  that  fu  is  a 
function  of  F  and  w,  and  that/v  is  a  function  of 

i  Tn  the  data  on  flume  traction  the  relation  of  capacity  to  fineness 
exhibits  peculiarities  quite  analogous  to  those  here  found  in  the  relation 
of  ti  to  fineness.  The  capacity  is  larger  for  very  fine  and  very  coarse 
d<5bris  than  for  intermediate  grades.  A  tentative  explanation  (see 
Chapter  XII)  connects  the  larger  capacity  for  fine  debris  with  a  tradi- 
tion in  process  from  traction  to  suspension. 


RELATION  OF  CAPACITY  TO  SLOPE. 


109 


S,  Q,  and  F.  Parallel  complexities  would  also 
arise  if  attempt  were  made  to  formulate  the 
relations  by  means  of  such  an  expression  as 


From  equation  (10)  (p.  96), 


The  sensitiveness  of  capacity  to  slope  appears 
to  be  a  function  of  the  conditions  jointly 
rather  than  severally. 

The  development  of  complexity  within  com- 
plexity suggests  that  the  actual  nature  of  the 
relation  is  too  involved  for  disentanglement  by 
empiric  methods,  but  that  conclusion  does  not 
necessarily  follow.  Just  as  a  highly  complex 
mathematical  expression  may  be  the  exact 
equivalent  of  a  fairly  simple  expression  of  a 
different  type,  so  a  physical  law  may  defy  for- 
mulation when  approached  in  a  certain  way 
yet  yield  readily  when  the  best  method  of 
approach  has  been  discovered. 

FORMULATION     WITH     CONSTANT     COEFFICIENT. 

For  the  relation  of  capacity  to  slope  the 
formula  equivalent  to  (34)  is 

<7=<VSV  ..(40) 


in  which  the  constant  coefficient  ct  is  the  value 
of  capacity  when  8=1. 


(41) 


Also  as  (10)  and  (40)  give  equivalent  expres- 
sions for  0,  we  may  equate  their  second  mem- 
bers: 

Cj^'-^OS-a)" 

Substituting  the  value  of  ct  from  (41), 


whence 
and 


S-a\n 


S1' 


_    log(S-g) -log  (! 
-n~  logS 


.(42) 


By  means  of  this  formula  the  values  of  j\  in 
Table  16  were  computed.  These  values  have 
been  subjected  to  systematic  comparison  with 
the  associated  conditions  (S,  Q,  w,  etc.)  in  the 
same  manner  as  were  the  values  of  the  ex- 
ponent \ ;  but  the  results  of  the  comparison 
need  not  be  given  in  detail  because  they  are 
parallel  to  those  for  ij.  They  are  in  fact  iden- 
tical so  far  as  verbal  statement  is  concerned 
and  differ  only  in  quantitative  ways.  It  is 
true  in  the  main,  and  almost  without  excep- 
tion, that  the  variations  of  j\  are  less  rapid 
than  those  of  i,;  and  their  average  range  is 
about  half  as  great. 


TABLE  16. —  Values  ofjlt  in  0=^8  h,  the  coefficient  c,  being  a  constant. 


Grade..! 


Conditions  of  experimentation  

w  

0.66 

1.00 

1.32 

1.96 

Q  

0.093 

0.182 

0.545 

0.182 

0.363 

0.734 

0.182 

0.363 

0.734 

0.363 

0.734 

0.119 

15 

39.4 

142 

37.6 

100 

231 

36.5 

104 

252 

96.1 

242 

359 

S 

Values  otji. 

0  2 

2.09 

1.91 

.4 

1.93 
1.88 
1.85 
1.83 

1.82 
1.81 
1.80 
1.80 

1.85 
1.82 
1.80 
1.79 

1.78 

1.86 
1.76 
1.72 
1.70 

1.68 
1.67 
1.66 
1.65 
1.65 

1.80 
1.75 
1.73 
1.71 

1.70 
1.68 

2.21 
2.03 
1.96 
1.90 

1.87 
1.84 
1.82 
1.81 

1.68 
1.61 
1.57 
1.55 

1.54 

1.39 
1.35 
1.33 
1.31 

.6. 

1.50 
1.49 
1.48 

1.47 

2.15 
2.10 
2.06 

2.04 
2.02 
2.01 
2.00 
1.99 

2.32 
2.23 
2.17 

2.13 
2.10 
2.08 
2.06 
2.05 

2.04 

.8 

2.06 
1.99 

1.95 
1.93 
1.91 
1  88 

1.0 

1.2 

2.36 
2.27 
2.20 
2  16 

I  4 

1.6 

1  8 

2  0 

2.12 
2.09 

1.80 

2  2 

1.99 

110 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 
TABLE  16. —  Values  ofji,  in  C=ctSh,  the  coefficient  c,  being  a  constant — Continued. 


Conditions  of  experimentation  

Grade.  . 

(B) 

to  

0.23 

0.44                                       0.66 

1.00 

«  

0.093 

0.182 

0.093 

0.182 

0.093 

0.182 

0.363 

0.545 

0.182 

0.363 

0.545 

0.734 

7.8 

13.4 

13.3 

28.9 

s 

Values  of  j\. 

0  2 

1.69 

1.95 

1.79 
1.68 
1.64 
1.62 
1.61 

1.61 
1.60 
1.59 
1.59 

4 

| 

1.% 

1.57 

1.78 

6 

.88 

1.89 

1.63 

1.53 

2.00 

1.73 

8                                         

4.57 
3.10 

2.60 
2.34 
2.18 
2.05 
1.96 

3.09 
2.49 

2.21 

2.05 
1.94 
1.86 

2.63 
2.31 

2.16 
2.02 
1.94 
1.88 
1.83 

.84 
.81 

.79 
.78 
.77 
.77 

2.47 
2.34 

2.26 
2.20 
2.16 
2.13 
2.10 

2.08 

1.85 
1.82 

1.81 
1.80 
1.79 
1.78 
1.77 

1.77 

.60 
.58 

.56 
.55 
.54 
.54 
.53 

1.50 
1.49 

1.48 
1.47 
1.46 
1.46 
1.45 

1.95 
1.92 

1.90 
1.88 
1.87 
1.86 
1.85 

1.84 
1.84 
1.83 

1.70 
1.68 

.66 
.65 
.64 
.64 
.64 

.63 

.63 

1.64 
1.62 

1.61 
1.61 
1.60 
1.60 
1.59 

1.59 

1.0                       

1.2                       

1.4            

1.6                

2  0 

2  2                       

1.89 

1.79 

24                                                  

1.76 

2.06 
2.04 

1.76 
1.76 

26                   

1.73 

2  g                                                

2.03 
2.02 

1.75 
1.75 

1.83 

3.0                                     

1.82 

32                                  

1.75 

3.4                

1.75 

3.6                              

1.74 

3.8            

1.74 

4.0  

1.74 

42                                

1  74 

Grade.. 

(B)                                                                                     (C) 

w  

1.32 

1.96 

0.44 

0.66 

Q 

0.182 

0.363 

0.545 

0.734 

0.363 

0.545 

0.734 

1.119 

0.093 

0.182 

0.093 

0.182 

29.2 

79.3 

140 

205 

67.8 

120 

189 

312 

10.9 

24.7 

13.8 

32.3 

S 

Values  of.;'i. 

0.2... 

1.96 

4  

2.26 
2.08 
2.00 
1.94 

.92 

.89 
.87 
.85 
.84 

1.92 
.82 
.78 
.75 

.72 
.71 
.70 
.69 
.68 

1.81 
1.76 
1.73 
1.71 

1.69 
1.68 
1.67 
1.67 

2.36 
2.16 
2.06 
2.01 

.97 
.94 
.92 
.90 
.89 

1.82 
1.74 
1.70 
1.66 

1.64 
1.63 
1.62 
1.61 
1.61 

1.69 
1.65 
1.62 
1.60 

1.58 

1.88 
.80 
.76 
.73 

.71 
.70 
.69 
.68 
.67 

.67 

.66 
.66 
.66 
.B5 

.65 
.65 
.64 
1.64 

6                     

1.79 
1.75 
1.73 

1.71 
1.70 
1.69 
1.68 

1.98 
1.91 
1.87 

1.84 
1.82 
LSI 
1.80 

3.08 
2.60 
2.39 

2.26 
2.17 
2.11 
2.06 
2.03 

1.99 

2.05 
.95 
.88 

.83 

.SO 
.77 
.76 
1.75 

1.74 

2.00 
.93 
.88 

.85 
.83 
.81 
.80 
.79 

.78 
.77 
.76 
.76 
1.75 

1.75 
1.75 
1.74 
1.74 
1.74 

1  73 

8 

1  0  

1  2... 

1  4                                       

1  6  
18.                





2.0  

2.2 

1.83 

1.82 

1.68 

1.88 

2.4... 

2.6  

1.82    '  

2.8  

1 



3.0  

1 

3.2... 

3.4.... 

3.6... 

3.8  

4.0  

4.2  

! 

i 

RELATION    OF   CAPACITY   TO   SLOPE. 
TABLE  16. —  Values  ofjlt  in  C=c,S'-'i,  the  coefficient  ct  being  a  constant — Continued. 


Ill 


Grade.  . 

(C) 

0.66 

1.00 

1.32 

Q 

0.363 

0.545       0.734       0.182       0.363 

0.545 

0.734       1.119 

0.182 

0.363 

0.545 

0.734 

73.4 

112           153          31.2         84.9 

140 

181          277 

21.3 

78.4 

130 

184 

s 

Values  of  /,. 

0.2... 

.76    ..                                 2 

.OB 

.73 

.  «1 
.Itt 

.at 

.57 

H 

IK 

1.65 

.86 
.60 
.54 
.49 
.47 

.45 
.44 
1.43 
1.42 
1.42 

1.41 

.4  

1.68 

.65    

2.12         1 
.98         1 
.91         1 
.87         1 

.84        1 
.82         1 
.80         1 

1.53         1.55 

1  92 

.6..                       

1.63 
1.61 
1.59 

1.58 

.61         1.58 
.59         1.58 
.58         1.57 

1.57   .. 

.51 
.49 
.47 

.46 
.45 
.44 

1.49          .51 
1.47           .49 
1.45           .48 

.44           .47 
.43           .47 
.43 

2.61 
2.46 
2.37 

2.32 
2.28 
2.25 
2.23 
2.21 

2.19 
2.17 

1.78 
1.71 
1.67 

1.64 
1.62 
1.60 
1.59 
1.58 

1.57 
1.56 

.62 
.58 
.55 

.53 

.51 
.50 
.49 
.49 

.48 
.47 

.8 

1.0  

1.2... 

1.4                                                   

1.57 

1.56 

1.6... 

1.56 

1.56    

1.8  

1.55 

1.55    79         1.54 

.43 

.42      . 

2.0  

2.2...                   
2.4                                   

1.55 

1.54 
1.54 

1.55    78         1.53 

1.55  L.                    .77         1.53 
.76         1.52 

.42 
1.42 

.42    
1.42    

2.6... 

1.54 

76         1.52 

2.8...   .                

.75         1.51 

Grade.. 

(C) 

CD) 

1.98 

0.66 

1.00 

Q 

0.363 

0.545 

0.734 

1.119 

0.093 

0.182 

0.545 

0.182 

0.363 

0.545 

0.734 

59.8 

111 

191 

345 

9.1 

29.5 

102 

24.0 

72.5 

108 

154 

S 

Values  of  j\. 

0.2  

.91 

.92 
.75 
.70 
.67 
.65 

.64 
.63 
.63 
1.62 

.4               

1.88 

2.02 
1.90 
1.84 
1.80 

1.78 
1.76 
1.74 
1.73 
1.72 

1.71 
1.71 

.74 
.69 
.66 
.64 

1.63 

1.62 
1.61 
1.61 
1.60 

1.60 

84 

.6                                                    .... 

2.38 
2.22 
2.14 

2.08 
2.04 
2.01 
1.99 
1.97 

1.95 

2.03 
.94 
.87 

.83 

.80 
.78 
1.76 
1.75 

1.74 

1.73 
1.66 
1.62 

1.60 
1.58 
1.56 

1.95 
1.89 
1.85 

1.82 

2.41 
2.29 
2.22 

2.18 
2.15 
2.12 
2.10 
2.09 

2.08 
2.07 

.77 
.72 
.69 

.67 
.65 
.64 
.63 
.63 

.62 
.61 

.94 
.90 
.87 

.85 
.84 
.84 
.83 
.82 

1.82 

.8... 

2.06 
2.01 

.98 
.95 
.93 
.91 
.90 

.89 

.88 
.87 

1.0                                                 

1.2 

1  4 

1.6..                                    

1  8 

2.0.                          

2.2... 

2.4 

2.6     . 

2.06 

1.70 

2.8  

.87 

Grade.. 

(D) 

(E) 

1.32 

0.66 

1.00 

1.32 

Q  

0.373 

0.734 

0.363 

0.734 

0.182 

0.363 

0.734 

1.119 

0.363 

0.734 

1.119 

58.9 

132 

24.8 

40.1 

14 

7 

33.6 

71.5 

139 

36.4 

72.6 

122 

S 

Values  of  j]. 

0.2... 

2.03 

1.76 
1.70 
1.68 
1.66 
1.65 

1.64 
1.64 
1.63 
1.63 
1.63 

.4 

2.02 

1.99 
1.94 
1.98 

1.89 
1.85 
1.82 
1.81 

1.80 
1.79 
1.79 
1.78 
1.78 

1.77 
1.77 

.6 

2.25 
2.16 
2.11 

2.07 
2.05 
2.03 
2.02 
2.01 

1.93 
1.88 
1.85 

1.83 

1.65 
1.64 
1.63 

1.63 
1.62 
1.62 
1.62 
1.62 

1.62 

.66 
.65 
.64 

.63 
.62 
.62 
.62 
.62 

1.56 
1.55 
1.54 

1.54 

1.82 
1.81 
1.81 

1.80 

.8                                               

1.0 

2.09 

2.06 
2.03 

2.17 

2.15 
2.13 
2.12 

.88 

.86 
.85 
.85 
.84 
.84 

.83 
.83 
89 

1  2 

1.4                            .                     

1.6 

1  8 

2.0                                                           

2.2                                                             

2.4 

1.62 

I 

2.6..                                                   

i 

2.8 

1 

8? 

1 

- 

112 


TRANSPORTATION    OP   DEBRIS  BY   RUNNING   WATER. 
TABLE  16. —  Values  of  jlt  in  C—cfih,  the  coefficient  c,,  being  a  constant — Continued. 


Conditions  of  experimentation  

Grade.  . 

(F) 

(G) 

10 

0.66 

1.00 

1.32 

0.66 

Q 

0.182 

0.363 

0.734 

0.182 

0.363 

a  734 

1.119 

0.363 

0.734 

1.119 

0.363 

Ci  

5.7 

20.4 

3S.7 

2.1 

13.9 

43.6 

79.6 

13.5 

40.3 

86.5 

8.3 

S 

Values  otji. 

0.2... 

.4  

.6... 

.8. 

2.85 
2.73 

2.65 
2.58 
2.52 
2.48 
2.45 

2.42 
2.39 
2.37 

2.14 
2.06 

2.02 
1.98 
1.95 
1.94 

1.95 
1.90 

1.87 
1.84 
1.82 
1.81 

2.52 
2.40 

2.32 

2.27 
2.22 

2.01 
1.94 

1.88 
1.85 
1.83 
1.80 

""S."  48 

3.21 

3.04 
2.91 
2.82 
2.76 

2.70 

2.65 
2.61 
2.58 
2.55 

1.0  

2.14 

2.08 
2.05 
2.03 
2.01 
1.99 

1.94 

1.91 

1.88 
1.86 
1.85 

1.2... 

2.67 
2.60 
1.53 
2.47 
2.43 

2.41 

3.87 
3.71 
3.59 
3.50 
3.43 

3.36 
3.31 
3.27 

2.54 

2.45 
2.38 
2.33 
2  29 

1.4... 

1.6  

1.8  

2.0 

2.2 

2  26 

2.4  

2.39 

2.6 

2.38 

2.8... 

3.0.  . 

Conditions  of  experimentation... 

Grade.. 

(G) 

(H) 

w 

0.66 

1.00 

1.32 

0.66 

Q  

0.734 

1.119 

0.363 

0.734 

1.119 

0.363 

0.734 

1.119 

0.363 

0.734 

1.119 

Ci  

30.3 

48.6 

4.8 

26.8 

60.1 

0.82 

19.0 

52.0 

0.76 

12.8 

25.2 

S 

Values  of  j\. 

0.2... 

.4  

.6  

2.77 

2.91 
2.60 
2.44 

2.34 
2.27 
2.22 
2.18 
2.15 

2.12 
2.10 

.8  

2.85 
2.65 

2.53 
2.45 
2.38 
2.34 
2.30 

2.27 
2.24 
2.22 

2.55 
2.43 

2.36 
2.31 
2.27 
2.24 
2.21 

2.19 
2.17 
2.16 

3.15 

2.88 

2.72 
2.61 
2.63 
2.48 
2.43 

2.39 
2.36 
2.34 

3.93 
3.43 

3.22 
3.00 
2.88 
2.79 
2.72 

2.66 
2.61 
2.57 
2.54 

2.81 
2.57 

2.44 
2.35 
2.27 
2.23 
2.19 

2.15 
2.12 
2.10 

4.68 
3.92 

3.54 
3.31 
3.15 
3.04 
2.95 

2.88 
2.82 
2.76 
2.72 
2  69 

3.63 
3.21 

2.98 
2.83 
2.75 
2.64 
2.58 

2.53 

2.49 
2.45 

1.0 

1.2... 

3.94 
3.67 
3.48 
3.35 
3.25 

3.16 
3.10 
3.04 
2.99 

7.65 
6.57 
5.93 
5.50 
5.20 

4.96 
4.77 
4.63 
4.49 

1.4... 

1.6.. 

1.8... 

5.34 
5.10 

4.92 

4.77 
4.65 
4.55 
4.45 
4.38 

2.0. 

2.2.. 

2.4  

2.6... 

2.8 

3.0  

2.94 

3.2  

2  66 

3.4  

4.32 

The  greatest  contrast  between  the  rates  of 
variation  of  the  two  exponents  is  found  when 
relations  to  slope  are  considered,  the  least 
when  relations  to  fineness  are  considered. 

The  variations  of  the  two  exponents  with 
respect  to  slope  are  illustrated  by  figure  35.  In 
the  example  from  which  the  curves  in  this 
figure  were  computed  the  range  of  variation, 
within  the  experimental  limits,  is  considerably 
above  the  average,  but  the  example  is  otherwise 
typical.  The  rate  of  change  is  everywhere 
smaller  for  j1  than  for  i,,  and  the  total  change, 
or  range  of  variation,  is  therefore  smaller.  This 
relation  prevails  throughout  the  range  of  slopes 
covered  by  the  experiments  but  would  not  be 


found  to  hold  for  slopes  far  outside  of  that 
range.  The  average  range  of  jlt  for  the  investi- 
gated cases,  falls  between  one-half  and  one- 
third  of  the  average  range  of  it. 

It  thus  appears  that  a  modification  of  the 
plan  of  formulation  which  dispenses  with  varia- 
tion in  the  coefficient  and  thereby  concentrates 
all  expression  of  variation  in  the  other  param- 
eter, the  exponent,  tends  also  to  diminish  di- 
versity in  that  parameter. 

EFFECT  OF  CHANGING  THE  UNIT  OF  SLOPE. 

The  values  of  jl  are  functions  not  only  of  the 
conditions  of  experimentation  and  of  the  con- 
stant slope  a  but  also  of  the  unit  used  for  the 


BELATION  OF  CAPACITY  TO  SLOPE. 


113 


measurement  of  slope.  (See  p.  99.)  In  the 
term  log  (1  —  a),  which  enters  into  the  value  of 
j,  (equation  42),  1  is  the  unit  of  slope;  and  the 
relative  magnitude  of  1  and  a  changes  as  the 
unit  changes.  Other  terms  of  the  formula  are 
also  (implicitly)  functions  of  the  unit,  but  the 
various  influences  are  not  compensatory,  and 
the  resultant  is  of  such  nature  that  the  values 
of  7,  vary  inversely  with  the  magnitude  of  the 
unit.  It  will  be  recalled  that  in  the  notation  of 
this  paper  the  symbol  S  pertains  to  the  unit 


distance 


'i'100) 


jjW  --     The  curve  marked  jt  in  figure  35  rep- 

resents values  of  the  exponent  computed  with 
use  of  the  smaller  unit,  and  the  curve  marked 
Tc  represents  a  coordinate  system  of  values  corn- 


Slope 


FIGURE  35.— Variations  of  exponents  d,  ;'i,  and  k,  in  relation  to  slope. 
The  scale  of  slope  is  in  per  cent. 

puted  with  use  of  the  larger  unit.  The  latter 
curve,  produced,  would  intersect  the  curve  of  ^ 
at  a  point  corresponding  to  s=  1  or  5=  100. 

It  would  be  possible,  by  employing  the  larger 
unit  in  the  notation  for  slope,  to  construct  a 
table  equivalent  to  Table  16  in  which  the  values 
of  7\  would  all  be  smaller  and  would  have  in  each 
series  less  range.  For  many  of  the  series  they 
would  approach  closely  the  associated  values  of 
n.  This  reduction  of  exponents  would  be  ac- 
companied by  an  enormous  increase  in  the 
values  of  coefficients,  each  value  of  ct  in  the 
table  being  magnified  by  the  factor  100".  The 
increase  would  result  from  the  fact  that  ct  is  the 
capacity  corresponding  to  unit  slope,  while  the 
unit  slope  in  that  case  would  be  45°.  Con- 
sidered as  capacities,  the  values  of  ct  would  be 
in  a  sense  fictitious,  because  the  laws  of  trac- 
tion wifh  which  we  are  dealing  do  not  apply 
(see  p.  63)  to  so  high  a  slope  as  45°. 

20921°— No.  86—14 8 


To  recur  for  a  moment  to  the  general  account 
of  the  index  of  relative  variation,  it  will  be 
recalled  that  the  index  was  shown  to  be  inde- 
pendent of  the  units  of  the  observational  quan- 

tities.    In  this  particular  instance  the  value  of 
o 

the  index,  n^  -  ,  has  two  factors,  of  which  the 

— 


first  is  an  abstract  number  and  the  second  is  a 
ratio  between  slopes  and  is  independent  of  the 
unit  of  slope.  The  ordinates  of  the  curve 
marked  \  in  figure  35  are  therefore  independent 
of  the  slope  unit,  and  the  curve  is  a  fact  of  ob- 
servation, plus  the  assumptions  of  the  formula 
of  adjustment. 

PRECISION. 

Because  the  values  of  \  were  computed  from 
values  of  n  and  a,  their  precisions  are  involved 
with  those  of  n  and  a,  but  the  relation  is  not 
simple.  The  precision  of  n  depends  partly  on 
the  harmony  of  the  observations  of  load  and 
slope  and  partly  on  the  precision  of  a.  It  is 
not  feasible  to  measure  the  precision  of  a  for 
individual  series  of  observations,  and  the  pre- 
cisions of  individual  values  of  n  and  it  are 
therefore  indeterminate.  All  that  has  been  at- 
tempted is  to  derive  a  rough  estimate  of  average 
precision  from  average  values  of  the  quantities 
involved.  The  estimated  average  probable 
error  of  values  of  n  is  3.9  per  cent,  and  the  cor- 
responding estimate  for  values  of  it  is  4.6  per 
cent. 

The  precision  of  it  varies  with  slope  within 
each  series  represented  by  a  column  in  Table 
15,  being  relatively  high  for  the  steeper  slopes. 
The  probable  error,  if  measured  in  the  same 
unit  with  i,,  is  much  larger  for  gentle  slopes 
than  for  steep  ;  if  measured  in  percentage,  it  is 
somewhat  larger  for  the  gentler  slopes.  Meas- 
ured in  percentage,  its  value  for  the  steeper 
slopes  approximates  the  corresponding  prob- 
able error  of  n. 

EVIDENCE      FROM     EXPERIMENTS     WITH     MIXED 
DEBRIS. 

The  observations  on  capacity  and  slope  when 
the  d6bris  transported  consisted  of  a  mixture 
of  two  or  more  grades  were  reduced  in  the  same 
general  manner  as  those  for  single  grades.  It 
was  not  thought  advisable  to  make  any  adjust- 
ment of  the  values  of  a,  but  each  logarithmic 
plot  was  treated  independently.  For  about 
one-third  of  the  mixtures  the  best  value 


114 


TRANSPORTATION    OF   DEBBIS  BY   RUNNING   WATER. 


appeared  to  be  zero.  For  another  it  fell  between 
the  values  which  had  been  assigned  to  the  finer 
and  coarser  components  of  the  mixture.  For 
the  remaining  third  it  exceeded  the  values  asso- 
ciated with  the  components.  When  the  load 
was  a  mixture  of  many  grades  combined  in  ap- 
proximately natural  proportions,  and  also  when 
it  consisted  of  unsorted  debris  from  a  river  bed, 


the  values  of  a  were  small  and  for  the  most  part 
zero. 

These  features  were  scrutinized  with  special 
interest  because  the  properties  of  grade  (E) 
had  suggested  that  great  range  of  fineness 
might  determine  very  low  values  of  a.  On  the 
whole,  the  data  from  mixtures  favor  that  view, 
but  their  support  is  by  no  means  unanimous. 


TABLE  17. —  Values  o/i,/or  mixtures  of  two  or  more  grades  of  debris. 


{Grade.  . 
w  

i.'oo 

0.363 

1.00         1.00         1.00         1.00         1.00 
0.363       0.363       0.363       0.363       0.363 

i.'oo1      i.'oo' 

0.363       0.363 

i.'oo' 

0.363 

(BiF2)     (B,F() 
1.00          1.00 
0.363       0.363 

1.00 
0.363 

g 

Parameters  of  interpolation  equation.  .  .•{*•  • 

0.25 
1.06 

0.30        0.20           000 
1.38         1.53         1.98         2.09         2.21 

0.25        0.25 
1.08        0.96 

0.50 
1.51 

0.50 
1.59 

0.50 
1.60 

0.30 
1.10 

S 

Values  otii. 

0.4  

1.82 
1.55 
1.42 

1.34 
1.30 
1.26 

2.76        2.30 
2.21         2.05 
1.97        1.92 

1.84          1.84 
1.76         1.79 
1.70   ... 

1.98            

.85         1.64 

2.20 
1.76 

1.57 

1.47 
1.40 
1.36 
1.32 

.8  

1  98 

.57         1.39 
.44          1.28 

.36         1.21 
.31         1.17 
1.14 

4.02 
3.02 

2.59 
2.35 
2.19 
2.09 

4.23 
3.17 

2.72 
2.47 
2.31 

4.27 
3.20 

2.75 
2.49 
2.33 

1.0 

1.98         2.09        2.21 

1.98        2.09        2.21 
1.98         2.09         2.21 
1.98         2.09        2.21 
2.21 

1.2 

1.4.. 

1.6  

1.8  '  

2.0  

2.21 

(Grade.. 

(CjEi) 
1.00 
0.182 

(CjE,) 
1.00 
0.363 

1.00 
0.182 

1.00 
0.363 

I'.OO 
0.182 

1.00 
0.363 

1.00 
0.182 

1.00 
0.363 

i!oo 

0.363 

i?oo' 

0.363 

1.00 
0.363 

IQ 

Parameters  of  interpolation  equation  ...<*" 

\n.  . 

0 
1.78 

0.30 
1.13 

0 
1.60 

0.40 
1.12 

0.20 
1.40 

0.30 
1.25 

0 
1.55 

0.30 
1.33 

0 
1.97 

0.30 
1.24 

0.50 
1.13 

S 

Values  of  >V 

0.4  

.6  

2.26 

1.97 
1.97 
1.97 

1.97 
1.97 
1.97 
1.97 

2.49 
1.99 
1.78 

1.66 
1.58 
1.53 
1.49 

3.01 
2.26 

1.93 

1.76 
1.64 

.8  

1.81 

2.23 

2.02 
1.79 

1.67 
1.60 
1.54 
1.51 
1.48 

1.45 

1.55 
1.55 

1.55 
1.55 
1.55 
1.55 
1.55 

1.55 

1.0  

1.62 

1.51 
1.44 
1.39 
1.36 

1.60 

.60 
.60 
.60 
.60 
.60 

1  60 

1.86 

.68 
.56 
.49 
.44 
.40 

1.75 

1.68 
1.63 
1.60 
1.57 
1.55 

1.54 

1.90 

1.77 
1.69 
1.63 
1.59 
1.56 

1.2        

1.4  

1.6     

1.78 
1.78 
1.78 

1  8 

2.0            

2.2        

1.78 

(Grade.. 
w 
Q:::::. 

1.00 
0.363 

1.00 
0.363 

(EjG,) 

1.00 
0.363 

1.00 
0.363 

(E,G2) 

1.00 
0.363 

(A,C,G,) 

1.00 
0.363 

(CDEFG) 

, 

Natural. 

1.00 
1.82 

1.00 
0.363 

1.00 
0.545 

1.00 
0.182 

1.00 
0.363 

Parameters  of  interpolatian  equation  ••-•{•" 

0.60 
1.47 

0 
1.87 

0.30 
1.90 

0.40 
1.28 

0 
2.03 

0.48            0 
1.67          1.63 

0 
1.48 

0 
1.22 

0.30 
1.43 

0 
1.84 

S 

Values  of  ii. 

0.4  

.84 

.84 
.84 
.84 

.84 
.84 
.84 

.6  

1.87 

22 

.8        .... 

1.87 
1.87 

1.87 

1.87 
1.87 
1.87 

3.04 
2.71 

2.53 
2.42 
2.34 

2.56 
2.13 

1.92 
1.79 
1.70 
1.64 

4.19 
3.22 

2.79 
2.55 
2.39 
2.28 

.63 
.63 

.63 
.63 
.63 
.63 

1.48 
1.48 

1.48 

1.48 
1.48 
1.4$ 

.22 
.22 

.22 
.22 
1.22 
1.22 

2.29 
2.04 

1.90 

1.82 
1.76 
1.72 

1  0 

3.68 

2.94 
2.58 
2.36 
2.21 

2.03 

2.03 
2.03 
2.03 
2.03 
2.03 

1  2 

1.4  ... 

1  6 

1.8  .       ... 

2.0  

2.4  

1  65 

2.6  

2.8  

! 

BELATION  OF  CAPACITY  TO  SLOPE. 


115 


The  adjusted  values  of  capacity  may  be 
found  in  Table  12,  together  with  a  variety  of 
other  data.  The  values  of  it  are  contained  in 
Table  17,  having  been  computed  for  the  range 
of  slopes  covered  by  the  observations.  In 
order  to  compare  these  with  the  indexes  of 
relative  variation  for  the  single  grades  from 
which  the  mixtures  were  made,  Table  18  has 
been  prepared,  containing  indexes  correspond- 
ing to  the  uniform  slope  of  1.2  per  cent.  An 


analysis  of  this  table  shows  that  3  mixtures 
are  more  sensitive  to  variation  of  slope  than 
are  components,  13  are  less  sensitive,  and  18 
show  sensitiveness  of  intermediate  rank.  A 
general  average  shows  the  mixtures  1 1  per  cent 
less  sensitive  than  the  (means  of)  components, 
but  the  contrast  is  much  more  pronounced  for 
natural  river  debris  and  for  the  most  complex 
artificial  combination  than  for  the  simpler 
mixtures. 


TABLE  18. — Comparison  of  values  of  i^  for  mixtures  and  their  components. 
[S-1.2;  w-l.OO.J 


Component  grades. 

a 

Value  of  it  when  ratio  of  finer  to  coarser  is  — 

Finer. 

Coarser. 

1:0 

4:1 

2:1 

1:  1 

1:2 

1:4 

0:  1 

(A)                                      

C)  .. 

0.363 
.363 
.363 
.182 
.363 
.363 
.363 
.363 
I        .182 
{        .363 
[        .545 
1        .182 
\        .363 

.81 
.81 
.65 
.81 
.55 
.55 
.79 
.34 
.81 
.55 
.44 
2.02 
1.81 

1.34 

1.55 
3.57 
2.54 
1.85 
1.79 
3.57 
3.57 
3.57 

(A)                .          

G)  

1.84 
1.36 

.84 
.21 
"  .78 
.51 
.66 
2.53 

al.98 
2.59 
ol.60 
1.68 
1.93 
1.92 
2  79 

o2.09 
2.72 
1.68 
1.67 
2.94 
02.03 

02.21 
2.75 
ol.55 
1.77 

(B) 

F) 

1C) 

E) 

(C) 

E)  

1.47 
01.97 
"1.87 

(C)                                  

G)  .. 

(E) 

G)                                 ... 

(  A\  ir\ 

(G) 

Mixture  of  (C),  (D),  (E),  (F), 
Natural  combination  ranging  i 

ind  (G)  

ol  63 

ol.48 

3.57 

ol  22 

1.90 

3.65 
2.54 

ol  84 

o  For  these, <r— 0  and  ii—n. 


The  general  tenor  of  the  evidence  from  mix- 
tures is  to  show  that  in  passing  from  the  labora- 
tory conditions  (with  graded  debris)  to  natural 
conditions  it  will  be  proper  to  reduce  the  esti- 
mates of  the  sensitiveness  of  capacity  to  changes 
of  slope. 

RELATION  OF  INDEX  TO  MODE  OF  TRACTION. 

One  of  the  possible  ways  of  bridging  the 
chasm  between  laboratory  conditions  and  river 
conditions  is  through  the  consideration  of 
modes  of  traction.  River  discharges  are  enor- 
mously greater  than  the  experimental,  river 
slopes  are  relatively  minute,  and  river  channel 
sections  are  very  dissimilar  in  form  and  pro- 
portions, but  the  three  modes  of  traction  have 
the  same  sequence  in  rivers  as  in  the  labora- 
tory troughs.  This  consideration  has  led  to  the 
separation  of  certain  data  connected  with  a  sin- 
gle mode  of  traction — the  one  characterized  by 
smooth  surfaces  of  water  and  debris.  Table  19 
contains  values  of  the  index  of  relative  varia- 
tion, I,,  associated  with  smooth  traction  and 
grouped  with  reference  to  various  conditions. 
In  the  upper  division  of  the  table  all  the  values 
pertain  to  the  same  grade  of  debris  (C)  and 


are  arranged  according  to  discharges  and  chan- 
nel widths;  in  the  lower  division  all  pertain  to 
the  same  width  of  channel  and  are  arranged  ac- 
cording to  discharges  and  grades  of  d6bris. 

TABLE  19. —  Values  of  \  associated  with  the  smooth  mode  of 
traction. 

I.  Values  pertaining  to  the  same  grade  of  debris,  (C). 


Width 
(feet). 

Value  of  »i  for  discharge  (ft.'/sec.)  of— 

0.093 

0.182 

0.363 

0.545 

0.734 

1.119 

0.44 
.66 
1.00 
1.32 
1.96 

2.15 
1.79 

1.88 
1.70 
1.81 
2.20 

1.59 
1.59 
1.61 
1.96 

1.59 
1.48 
1.55 
1.79 

1.58 
1.48 
1.51 
1.62 

1.93 

II.  Values  pertaining  to  the  same  width  of  channel,  0.66  feet. 


Grade. 

Value  of  ii  for  discharge  (ft.'/see.)  of— 

0.093 

0.182 

0.363 

0.545 

0.734 

1.119 

(B) 

1 

(EJ 
(G) 
H$ 

2.09 
1.79 
2.01 

1.82 
1.70 
1.72 
1.99 

1.61 
1.59 

1.54 
1.59 
1.64 

1.58 

2.13 
2  41 

2.19 
2.65 

2.20 
2.54 

3.35 

116 


TRANSPORTATION    OF   DEBRIS  BY   RUNNING   WATER. 


Inspection  of  the  table  shows  that  under 
this  special  condition  the  index  (1)  varies  in- 
versely with  discharge,  (2)  varies  inversely 
with  fineness,  and  (3)  varies  both  inversely 
and  directly  with  width.  In  all  these  respects 
the  variations  are  similar  to  those  noted  when 
the  constant  condition  is  slope,  but  the  rates  of 
variation  are  not  the  same.  To  illustrate  the 
differences  in  rate  I  introduce  Table  20,  in 
which  data  of  Table  19  are  compared,  in  par- 
allel columns,  with  similar  data  (from  Table 
15)  conditioned  by  constant  slope.  The  same 
data  are  also  presented  graphically  and  with 
some  generalization  in  figure  36.  On  giving 
attention  to  the  first  division  of  the  table  and 
to  the  upper  diagram,  it  will  be  seen  that  the 
variation  of  the  index  with  discharge  is  some- 
what less  when  the  constant  factor  is  mode  of 
traction  than  when  it  is  slope.  The  second 
division  and  second  diagram  show  that  the 
variation  with  fineness  is  also  less  for  constant 
mode  of  traction.  The  third  division  and 
diagram  indicate  that  the  variation  with  width 
of  channel  is  of  the  same  order  of  magnitude  in 
the  two  cases,  but  that  for  constant  mode  of 
traction  the  minimum  value  of  the  index  is 
associated  with  greater  width. 


The  comparison  does  not  indicate  that  the 
condition    of    uniform    mode    of    traction    is 


FIGURE  36.— Variations  of  >i  under  condition  of  uniform  mode  of  trac- 
tion (  T,  T,  T),  and  under  condition  of  uniform  slope  (S,  S,  S).  Ver- 
tical distances  represent  i\,  with  a  common  scale  but  different  zeros. 
Horizontal  distances  represent,  for  the  upper  pair  of  curves,  discharge; 
for  the  second  pair,  logarithms  of  fineness;  for  the  lower  pair,  width  of 
channel;  the  values  Increasing  from  left  to  right. 

greatly  to  be  preferred  to  that  of  uniform  slope 
as  a  basis  for  the  extension  of  laboratory  gen- 
eralizations to  large  natural  streams. 


TABLE  20. — Relations  o/i,  to  discharge,  debris  grade,  and  channel  width,  when  conditioned  (1)  by  a  constant  mode  of  traction 

and  (2)  by  a  constant  slope. 


Grade,  (C) 
w,  0.66  foot. 

w,  0.66  foot. 
Q,  0.363  ft.s/sec. 

Grade,  (C) 
Q,  0.182  ft.'/sec. 

Q 

Values  of  n  with— 

Grade. 

Values  of  ii  with— 

w 

Values  of  ii  with  — 

Smooth 
traction. 

Constant 
slope  1.0 
per  cent. 

Smooth 
traction. 

Constant 
slope  1.4 
per  cent. 

Smooth 
traction. 

Constant 
slope  1.2 
per  cent. 

0.093 
.182 
.363 
.545 
.734 

1.79 
1.70 
1.59 
1.59 
1.58 

1.87 
1.73 
1.59 
1.58 
1.57 

(B) 

(C) 
(E) 
(G) 
H) 

1.61 
1.59 
2.13 
2.41 
3.35 

1.54 
1.55 
1.99 
2.71 
4.69 

0.44 
.66 
1.00 
1.32 

1.88 
1.70 
1.81 
2.20 

1.81 
1.70 
1.81 
2.27 

IN  CHANNELS  OP  SIMILAR  SECTION. 

THE    CONDITIONS. 

It  will  be  shown  in  the  following  chapter  that 
one  of  the  important  conditions  affecting  capac- 
ity is  the  relation  of  stream  depth  to  stream 
width,  or  the  form  ratio  E.  The  matter  has,  in 
fact,  already  received  some  attention  in  con- 
nection with  the  variation  of  a.  Now,  in  each 
observational  series  the  width  is  constant  while 
the  depth  varies,  so  that  the  form  ratio  is  a 
variable.  Its  variations  accompany  and  are 


inseparable  from  those  of  slope;  and  the  varia- 
tion of  capacity  (within  an  observational 
series),  which  up  to  this  point  has  been  treated 
as  if  it  were  purely  a  function  of  slope,  is  in 
reality  a  function  of  slope  and  form  ratio 
jointly.  To  separate  the  two  factors  and  there- 
by discover  the  relation  of  capacity  to  slope  for 
streams  of  similar  section,  it  is  necessary  to 
bring  together  data  obtained  by  use  of  troughs 
with  different  widths,  selecting  points  of  two 
or  more  adjusted  series  which  are  characterized 
by  the  same  ratio  of  depth  to  width.  In  every 


RELATION  OF  CAPACITY  TO  SLOPE. 


117 


such  comparison  the  capacity  and  slope  asso- 
ciated with  the  narrower  trough  are  relatively 
large,  while  for  the  wider  trough  they  are  rela- 
tively small.  The  opportunities  for  comparison 
are  not  abundant,  because  in  the  main  the 
observational  series  which  are  of  like  conditions 
except  as  to  width  do  not  overlap  in  respect 
to  form  ratio. 

SIGMA    AND   THE    INDEX. 

In  the  records  of  the  main  body  of  experi- 
ments 24  cases  of  overlap  are  found,  all  asso- 
ciated with  the  finer  grades  of  debris,  from  (A) 
to  (D).  In  each  of  these  the  comparison  in- 
cludes two  widths  only,  no  instance  occurring 
in  which  it  can  be  extended  to  three.  There 
is,  however,  a  special  group  of  experimental 


series,  planned  in  part  for  this  particular  pur- 
pose, in  which  the  trough-width  interval  is  so 
small  that  triple  overlaps  occur.  The  special 
experiments  were  made  with  debris  of  grade 
(C);  the  trough  widths  were  1.0,  1.2,  1.4,  1.6, 
1.8,  and  1.96  feet;  and  the  experiments  yield 
nine  triple  overlaps.  In  six  of  these  the  extent 
of  overlap  is  such  that  numerical  comparisons 
have  been  made  for  more  than  one  value  of  R. 
With  the  aid  of  the  computation  sheets 
described  on  page  95,  a  table  was  compiled  in 
which  adjusted  capacities  and  slopes  were 
arranged  with  respect  to  form  ratio,  and  in 
this  table,  which  has  not  been  printed,  the 
matter  of  overlaps  was  canvassed.  Table  21 
contains  the  data  involved  in  the  triple 
overlaps. 


TABLE  21. — Selected  data,  for  grade  (C),  shewing  the  relation  of  capacity  to  slope  when  the  form  ratio  is  constant. 


Q-  0.734 

Q-0.923 

Q-1.021 

Q-1.119 

1C 

S 

C 

w 

S 

C 

w 

S 

C 

1C 

S 

C 

0.07  

.no 

1.38 

354 

.SO 

1.17 

262 

.96 

.75 

118 

.08  

.60 

1.08 

226 

.HO 

.89 

151 

.96 

.49 

48 

.09  

1.60 

1.29 

413 

1.80 

.94 

210 

1.96 

.58 

88 

.10 

1.40 

1.33 

346 

1.60 

1.19 

405 

1.00 

.72 

102 

1.80 

.82 

195 

1.80 

.55 

57 

1.96 

.50 

75 

.11 

1.40 

1.06 

224 

1.40 

1.44 

522 

1.60 

1.07 

380 

1.60 

.60 

71 

1.60 

.83 

185 

1.80 

.74 

193 

1.80 

.45 

35 

1.80 

.60 

83 

1.96 

.50 

88 

.12  .. 

1.40 

1.17 

357 

1.60 

.87 

261 

1.60 

.69 

130 

1  SO 

62 

134 

1.80 

.50 

56 

1.96 

.41 

55 

.13... 

1.20 

1.43 

382 

1.40 

.96 

243 

1.40 

1.16 

380 

1.40 

.72 

103 

1.60 

.58 

94 

1.60 

.65 

135 

1.60 

.44 

37 

1.80 

.42 

36 

1.80 

.47 

58 

.14  

1.40 

1.20 

470 

1.60 

.60 

130 

1.80 

.45 

66 

.15 

1  40 

99 

331 

1.60 

.51 

95 

1  80 

39 

.16  

1.40 

.83 

236 

1.60 

.44 

71 

1.80 

.34 

To  illustrate  the  use  made  of  such  data,  the 
case  of  #=1.119  ft.3/sec.  and  5  =  0.15  is 
selected.  In  that  example  the  values  of 
slope  and  capacity  are  given  for  the  trough 
widths  1.4,  1.6,  and  1.8  feet.  These  values 
come  from  three  adjusting  equations,  which  are, 
for  the  widths  severally,  £=386  (S-0.08)1-72, 
(7=400  (S-O.IO)1-60,  and  (7=430  (S-0.12)  1-67. 
The  graphs  of  the  equations  are  shown  in 


figure  37.  On  each  graph  is  a  dot  indicating 
the  point  which  corresponds  to  the  tabulated 
values  of  C  and  S,  and  for  each  of  these  points 
the  ratio  R  is  0.15.  The  curves  represent  the 
relations  of  C  to  S  under  the  condition  of 
uniform  width.  The  three  dots  are  points  on 
an  undrawn  curve  to  express  the  relation  of 
C  to  S  under  the  condition  of  constant  form 
ratio. 


118 


TRANSPORTATION    OF   DEBRIS   BY   RUNNING   WATER. 


It  is  convenient,  in  discussing  this  undrawn 
curve,  to  assume  that  its  equation  involves  a 
and  is  otherwise  of  the  same  type  as  the  equa- 
tions used  in  discussing  the  data  for  constant 
width.  In  this  case,  moreover,  the  assump- 
tion is  countenanced  by  the  fact  that  traction 
is  limited  by  the  competent  slope.  By  making 
the  assumption,  it  is  possible  to  compute  all 
the  parameters  of  the  curve  from  the  coordi- 
nates of  the  three  known  points  and  write  its 
equation : 

£=469  (S -0.23) '-25      (43) 

In  the  present  connection  the  most  significant 
of  the  parameters  is  o\  and  the  values  of  a 


have  been  computed  for  each  of  the  cases  of 
Table  21.     They  are  assembled  in  Table  22. 

TABLE  22. —  Values  of  a  corresponding  to  data  in  Table  SI. 


R 

Value  of  »  when  Q  is— 

0.734 

0.923 

1.021 

1.119 

0.07... 
.08... 
.09 

f            ° 

\         o 

—.44 

j""+."36" 
{      +.33 
+.31 

+.04 

.  10.  .  . 
.11... 
.12... 
.13... 
.14... 

/       +.20 
\      +.20 
........ 

/"+'."  6s" 

\    +.11 

""+."i3" 

I     +.21 
<     +.24 
1    +.24 

.15  . 

16 

400 


a 
to 
<J 


Slope 


FIGUHE  37.— Curves  of  C— /(S)  for  trough  widths  of  1.4, 1.6,  and  1.8  feet,  showing  points  which  agree  as  to  form  ratio. 

Grade  (C).    Q-1.119ft.3/sec.    fi-0.15. 


One  value  of  a  has  a  negative  sign.  As  this 
yields  positive  capacity  for  negative  slope, 
thus  transcending  the  physical  conditions  of 
traction,  and  as  the  value  is  numerically  large, 
an  error  of  importance  is  betrayed.  The 
error  appears  not  to  be  one  of  computation 
and  must  be  ascribed  either  to  the  observa- 
tional data  or  to  assumptions  made  in  the 
work  of  reduction  and  adjustment.  In  either 
case  its  existence  serves  to  qualify  the  values 
of  a  as  of  low  precision. 


In  view  of  that  qualification,  the  study  of  the 
values  of  a  in  detail  appears  unprofitable  and 
only  their  mean  will  be  considered.  In  deriv- 
ing the  general  mean  a  subsidiary  mean  was 
first  found  for  each  case  of  overlap  and  these 
were  afterward  combined,  this  course  being 
followed  because  when  several  values  of  a  are 
derived  from  the  same  overlap  of  series  the}* 
are  not  independent.  The  general  mean  thus 
derived  is  +0.08,  and  this  may  be  compared 
with  a  mean  similarly  obtained  from  the  asso- 


BELATION  OF  CAPACITY  TO  SLOPE. 


119 


elated  equations  under  condition  of  constant 
width,  namely,  +0.12.  So  far  as  this  moder- 
ate difference  in  magnitude  has  significance,  it 
indicates  that  the  index  of  relative  variation,  it, 
has  the  smaller  range  under  the  condition  of 
constant  form  ratio;  for  the  index's  range  in 
magnitude  is  a  direct  function  of  a. 

THE    SYNTHETIC    INDEX. 

Another  mode  of  viewing  the  data  of  Table 
21  ignores  the  middle  term  of  the  triple  over- 
lap and  considers  only  the  first  and  last.  From 


the  values  of  capacity  and  slope  in  these  two 
terms  a  value  of  the  synthetic  index  can  be 
computed  by  equation  (35).  It  is  possible  in 
this  way  to  broaden  the  range  of  data  by  in- 
cluding the  cases  of  simple  overlap;  and  it  is 
possible  to  make  many  pertinent  comparisons 
by  computing  the  corresponding  values  of  the 
synthetic  index  for  observational  series  in 
which  the  constant  condition  is  width  instead 
of  form  ratio.  The  indexes  with  form  ratio 
constant  and  with  width  constant  may  be 
symbolized  severally  by  7*  and  Iw. 


TABLE  23. —  Values  of  synthetic  index  under  condition  that  R  is  constant.  (Is);  u-ith  coordinate  mines,  under  condition  that 

w  is  constant,  (/tt). 


Grade. 

Q. 

Width. 

K. 

/,. 

I*. 

Channel 
narrow. 

Channel 
wide. 

(A) 
(B) 

(C) 

(D) 

Ad 

0.363 
.734 
.182 

.363 
.545 

.734 
.734 
.182 
.182 

.182 
.363 

.365 
.545 

.734 
.734 

.734 
.734 
.734 

.734 
.923 

.923 
1.021 
1.021 
1.119 

1.119 

.363 
.734 

usted  mea 

1.32 
1.32 
1.00 

.66 
1.32 

1.00 
1.32 
.44 
.66 

1.00 

.66 
1.00 

1.32 
1.00 

.66 
1.00 

1.20 
1.32 
1.40 

1.60 
1.40 

1.60 
1.40 
1.60 
1.40 

1.60 

1.00 
1.00 

us 

1.96 
1.% 
1.32 

1.00 
1.96 

1.32 
1.96 
.66 
1.00 

1.32 

1.00 
1.32 

1.96 
1.32 

1.00 
1.32 

1.60 
1.96 
1.80 

1.96 

1.80 

1.96 
1.80 
1.96 
1.80 

1.% 

1.32 
1.32 

.06 
.07 
.10 
.12 
.06 
.07 
.08 
.09 
.20 
.07 
.08 
.20 
.12 
.30 
.10 
.12 
.14 
.05 
.06 
.07 
.20 
.30 
.10 
.12 
.06 
.12 
.14 
.40 
.16 
.18 
.20 
.13 
.09 
.10 
.11 
.07 
.08 
.11 
.12 
.13 
.09 
.13 
.10 
.14 
.15 
.16 
.11 
.12 
.10 
.16 
.18 
.20 

2.01 
2.23 
1.77 
1.90 
1.95 
2.09 
2.22 
2.49 
1.69 
2.00 
2.11 
1.80 
2.05 
1.71 
1.74 
1.88 
2.02 
2.11 
2.36 
2.67 
.53 
.96 
.72 
.87 
.22 
.56 
.64 
.46 
1.47 
1.54 
1.59 
2.01 
1.60 
2.06 
2.17 
1.81 
1.96 
2.10 
2.18 
2.30 
1.93 
2.08 
1.95 
2.00 
2.10 
2.20 
1.93 
2.07 
2.02 
1.86 
1.94 
2.11 

1.88 

1.74 

1.80 
1.82 

1.99 
2.17 
1.75 

1.87 
2.00 
2.06 

1.92 
2.02 
2.16 

1.75 
1.96 

1.75 

1.69 

1.82 

1.88 
1.79 
1.92 
2.04 

.69 
.75 
.82 
.70 
.77 
.87 
.62 

2.26 
2.43 
1.64 
2.07 
1.66 
1.88 
2.24 
1.52 

.58 
.65 
.72 
.44 
.48 
.20 
.48 
.52 
.73 
.91 

1.60 

.00 

2.01 

.92 
.98 

1.95 
2.07 

.91 

2.08 

.86 
.99 
1.86 

1.98 

1.99 

1.49 

2.13 

.87 
.94 
.67 

2.07 
2.13 

.66 
.72 

1.97 

1.72 

1.89 

Table  23  contains  values  of  7*  computed  for 
the  9  cases  of  overlap  shown  in  Table  21  and 
for  24  cases  of  simple  overlap  previously  men- 
tioned. Where  the  overlap  is  extensive,  values 
of  Is  were  computed  for  two  or  more  values 


of  R.  Each  of  these  values  is  based  on  data  of 
two  width-constant  series,  pertaining  to  differ- 
ent trough  widths,  and  whenever  the  extent 
of  such  an  associated  width-constant  series  per- 
mitted, a  value  of  Iw  was  computed  from  its 


'120 


TRANSPORTATION    OF   DEBEIS  BY   RUNNING   WATER. 


data  for  the  same  range  of  slope  as  that  covered 
by  the  value  of  Is.  The  table  thus  comprises 
data  for  the  coordinate  values  of  the  indexes 
under  conditions  of  constant  form  ratio  and 
constant  width. 

The  most  important  generalization  from  the 
data  in  Table  23  is  contained  in  the  adjusted 
means  which  appear  at  the  bottom.  In  the 
derivation  of  these  means  due  account  was 
taken  of  the  fact  that  the  several  individual 
values  from  the  same  overlap  are  not  inde- 
pendent and  also  of  the  fact  that  the  coordinate 
data  for  constant  width  are  incomplete.  The 
mean  for  constant  form  ratio,  1.88,  is  practi- 
cally the  same  as  that  one  for  constant  width 
which  is  associated  with  the  wider  channels, 
1 .89,  and  is  notably  greater  than  the  mean  as- 
sociated with  the  narrower  channels,  1 .72.  On 
the  whole  it  is  indicated  that  the  sensitiveness 
'of  capacity  to  slope  is  the  greater  for  traction 
conditioned  by  constant  form  ratio,  in  the  pro- 
portion of  1.88  to  1.805,  or  as  1.04  to  1. 

When  the  values  of  Is  and  /„,  are  arranged 
according  to  the  associated  values  of  E,  the  fol- 
lowing relations  are  brought  out : 


Mea 

i  In. 

1& 

B 

IM- 

Narrower 
trough. 

Wider 
trough. 

~fai 

0.05  to  0.08... 

2.14 

1  86 

2  10 

1  08 

.  09  to    .12 

1  93 

1  gi 

1  94 

1  03 

.14  to   .40  

1.80 

1  63 

1  86 

1  03 

SUMMARY. 

Only  a  small  fraction  of  the  observational 
data  are  available  for  the  discussion  of  the 
capacity-slope  relation  under  the  condition  of 
constant  form  ratio,  and  the  discussion  is  there- 
fore limited  to  a  comparison  of  its  features  un- 
der that  condition  with  corresponding  features 
under  the  condition  of  constant  channel  width. 
The  results  of  such  comparison  may  be  sum- 
marized as  follows:  The  sensitiveness  of  trac- 
tional  capacity  to  variation  of  slope  is  in  gen- 
eral greater  under  the  condition  of  constant 
form  ratio,  but  the  difference  is  of  moderate 
amount.  The  difference  is  somewhat  less  (at 
least  within  the  limits  of  available  data)  for 
broad  and  shallow  streams  than  for  streams 
that  are  narrow  and  deep.  The  range  of  sensi- 
tiveness, or  its  variation  with  variation  of  slope, 
appears  to  be  somewhat  less  under  the  condi- 
tion of  constant  form  ratio.  The  generaliza- 


tions in  regard  to  traction  by  currents  of  varia- 
ble depth  but  invariable  width  may  be  ex- 
tended, with  only  moderate  qualification,  to  the 
case  of  currents  which  retain  geometric  simi- 
larity of  section  while  slope  is  varied. 

REVIEW. 

With  increase  of  the  slope  of  descent  goes  in- 
crease of  a  stream's  energy  (per  unit  time,  per 
unit  distance).  With  the  increase  of  energy 
goes  increase  of  capacity  for  the  transportation 
of  ddbris  along  the  channel  bed.  The  increase 
of  energy  is  strictly  proportional  to  the  increase 
of  slope,  but  the  increase  of  capacity  follows  a 
different  law.  The  law  is  not  simple,  but  one 
feature  persists  through  all  its  manifestations: 
The  capacity  for  traction  increases  more  rap- 
idly than  the  slope.  The  difference  in  rapidity, 
or  the  magnitude  of  the  difference  between  the 
rates  of  change  for  capacity  and  slope,  is  itself 
a  variable,  depending  on  a  variety  of  condi- 
tions. The  study  of  the  relation  of  capacity  to 
slope  is  here  treated  as  a  study  of  the  influence 
of  conditions  on  the  magnitude  of  the  differ- 
ence between  the  two  rates  of  change. 

The  magnitude  of  that  difference  is  indicated 
by  a  quantity,  of  the  nature  of  an  exponent, 
called  the  index  of  relative  variation  (of  capac- 
ity, as  compared  to  slope),  and  designated  by 
it.  The  index  may  be  defined  as  the  first 
differential  coefficient  of  log  0  with  respect  to 
log  S.  It  is  illustrated  by  saying  that  the 
capacity  varies,  instantaneously,  as  the  i1 
power  of  the  slope. 

For  the  greater  part  of  the  field  covered  by 
the  experiments  the  index  falls  between  1.4 
and  3.0,  but  under  some  conditions  it  is  con- 
siderably higher.  It  varies  with  slope,  being 
higher  for  low  slopes  and  lower  for  high.  It 
varies  with  discharge,  being  relatively  high  for 
small  discharges.  It  varies  with  fineness,  being 
relatively  high  for  coarse  debris.  Briefly,  it 
varies  inversely  with  slope,  discharge,  and 
fineness.  It  varies  also  with  width  of  channel, 
decreasingly  for  relatively  narrow  channels  and 
increasingly  for  relatively  broad  channels;  so 
that,  for  any  particular  combination  of  slope, 
discharge,  and  fineness  there  is  a  width  charac- 
terized by  a  minimum  value  of  the  index. 

It  is  furthermore  true  that  no  one  of  these 
variations  is  itself  constant  in  rate,  the  rate  of 
each  having  its  own  law  of  variation.  Thus 
the  complexity  of  the  relation  of  capacity  to 


RELATION    OF   CAPACITY   TO   SLOPE. 


121 


slope  is  such  as  to  be  characterized  by  varia- 
tions of  the  rates  of  variation  of  rates  of  varia- 
tion; and  there  are  even  vistas  of  higher  orders 
of  variability.  The  law  connecting  capacity 
with  slope  may  be  susceptible  of  much  more 
compact  expression,  but  such  formulation  must 
probably  await  the  development  of  a  mechan- 
ical theory  of  stream  traction. 

Formulation  founded  on  the  index  of  rela- 
tive variation,  while  bringing  out  clearly  cer- 
tain general  features  of  the  law,  is  not  able  to 
afford  a  complete  quantitative  statement.  It 
may  be  likened  to  a  map  in  definite  hachures 
as  contrasted  with  one  in  definite  contours. 
As  the  hachure  tells  the  direction  and  rate  of 
slope  but  omits  the  absolute  altitude,  so  the 
index  tells  the  relative  change  under  given  con- 
ditions but  omits  the  absolute  capacity.  To 
remedy  this  defect,  the  experiment  was  tried 
of  substituting  for  the  equation  C=vlSl>,  in 
which  vl  is  variable,  the  equation  C=clSil,  in 
which  Cj  is  constant  for  each  series  of  observa- 
tions. Formulation  by  means  of  ct  and  jl  is 
more  nearly  analogous  to  the  contour  map, 
but  the  variability  of  the  exponent  jt  is  no  less 
formidable  than  that  of  \  while  the  definition 
and  derivation  of  y\  are  less  simple  and  its 
significance  is  less  clear. 

Further  utilizing  the  analogy  of  the  map,  we 
may  think  of  the  capacity-slope  relation  as  an 
undulating  topography,  in  which  the  vertical 

0         f(C) 
element  is  -^  or    s,c,\  and  the  horizontal  ele- 


ments  are  qualifying  conditions.  Formulation 
is  a  mode  of  representing  this  topography,  the 
hills  and  valleys  of  which  do  not  depend  on  the 
mode  but  are  real.  Two  modes  have  been 
tried,  each  with  limitations,  but  the  ideal  mode 
is  not  known.  The  contour  map  or  the  relief 
model  would  serve  admirably  if  the  qualifying 
conditions  were  two  only,  but  as  they  number 
at  least  four,  a  graphic  or  plastic  expression  is 
possible  only  in  space  of  n  dimensions. 

By  reason  of  the  complexity  of  the  relation 
of  capacity  to  slope  and  because  of  the  lack  of 
a  mechanical  theory  of  flow  and  traction,  the 
laboratory  data  do  not  warrant  inferences  as  to 
the  quantitative  relations  of  capacity  to  slope 
for  rivers. 

Of  various  attempts  to  evade  the  complexity, 
two  are  thought  worthy  of  record.  In  each 
series  of  experiments  the  mode  of  traction 
changes  with  increase  of  slope,  first  from  dune 


to  smooth,  then  from  smooth  to  antidune,  but 
the  critical  slopes  are  not  the  same  for  different 
discharges  or  widths  or  degrees  of  fineness.  It 
appeared  possible  to  gain  in  simplicity  by 
treating  separately  the  data  associated  with  a 
single  mode  of  traction,  and  data  for  the 
smooth  mode  were  accordingly  segregated  and 
discussed.  Greater  simplicity  was  not  found, 
but  the  range  of  variation  is  somewhat  smaller 
for  the  single  mode  of  traction. 

The  second  attempt  was  connected  with  the 
form  of  cross  section  of  the  current.  Within  a 
single  series  of  experiments  the  width  was  con- 
stant and  the  depth  varied,  so  that  the  capacity 
was  conditioned  not  only  by  slope  but  by  form 

ratio,  R  =  ~.  By  comparing  one  observational 

series  with  another  it  is  possible  to  obtain  data 
conditioned  by  difference  in  slope,  but  without 
difference  in  form  ratio.  The  discussion  of 
such  data  developed  only  moderate  modifica- 
tion of  the  results  previously  obtained  and  no 
reduction  in  complexity. 

DUTY  AND  EFFICIENCY. 

For  the  purposes  of  this  paper  duty  has  been 

defined  as  the  ratio  of  capacity  to  discharge: 

n 
U=  -Q.    Combining  this  with  C=f(S)  ,  the  most 

general  expression  for  the  relation  of  capacity 


to  slope,  we  have  U= 


Under  no  form  dis- 


covered for/(£)  is  this  expression  reducible  to 
simpler  terms.  For  each  value  of  discharge, 
duty  is  simply  proportional  to  capacity;  and 
the  entire  discussion  of  this  chapter  applies  to 
duty  as  well  as  capacity.  The  parameters,  n, 
\,  ji,  !«,,  and  a  may  be  transferred,  without 
modification,  to  formulas  for  duty. 

Efficiency  has  been  defined  as  the  ratio  of 
capacity  to  the  product  of  discharge  by  slope: 


-(44) 


The  combination  of  this  with 
C=v1S<i. 


yields 


--.(36) 

=  v  —  =  v-iS'>-1  .(45) 


The  transformation  of  the  exponent  is  im- 
portant.    While  capacity  varies  as  the  it  power 


122 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


of  slope,  efficiency  varies  as  the  i,  —  1  power. 
The  index  of  relative  variation  is  1  less  for 
efficiency  than  for  capacity.  Table  15  could 
therefore  be  adapted  to  efficiency  by  diminish- 
ing all  its  values  by  unity.  As  the  lowest 
known  values  of  \  are  greater  than  unity — the 
lowest  in  Table  15  is  1.31— it  follows  that  effi- 
ciency is  an  increasing  function  of  slope  under 
all  tested  conditions. 

The  combination  of  (44)  with 


vields 


-(40) 
-(46) 


showing  that  in  passing  from  the  field  of  ca- 
pacity to  that  of  efficiency  the  exponent  asso- 
ciated with  a  constant  coefficient  also  is  reduced 
by  unity. 

Following  the  form  of  equation  (35),  we  have 

log  c"- log  a          ,35 , 

7'  =  logS"-logS'--- 

where  C'  and  C"  are  specific  values  of  capacity 
corresponding  severally  to  the  slopes  S'  and  S". 
Designating  by  Ie  the  synthetic  index  of  effi- 
ciency in  relation  to  slope,  and  by  E'  and 
E"  the  efficiencies  corresponding  to  C'  and 
C" ,  we  have 


"" 


log  ff"-l 


E' 


log  S' '-log  S' 


.(35b) 


As  C=  ESQ,  log  C' = log  E'  +  log  S'  +  log  Q,  and 
log  C"'  =  log  E"  +log  S"  +log  Q.  Substituting 
these  values  in  (35a)  and  reducing,  we  have 


Subtracting  the  members   of  this   expression 
from  those  of  (35b)  and  transposing,  we  have 

/.-/,-!-  (47) 


That  is,  the  synthetic  index  of  relative  varia- 
tion of  efficiency  with  reference  to  slope  is  less 
by  unity  than  the  corresponding  index  for 
capacity. 

It  is  evident  that  at  competent  slope,  when 
capacity  is  zero,  efficiency  also  is  zero.  Like 
capacity,  it  increases  with  increase  of  slope. 
Under  the  assumption  that  its  law  of  increase 
is  of  the  same  type,  its  value  varying  with  a 


power  of  (S  —  a),  two  expressions  have  been 
derived,  but  neither  has  been  found  reducible 
to  simple  form.  The  algebraic  work  being 
omitted,  they  are 


n  —  ,    — ?W— 


S-ir 


logS-2 


log  (S-»)-2 

In  the  first  of  these  expressions  the  coeffi- 
cient is  variable,  being  a  function  of  S;  so  that 
the  exponent  may  be  regarded  as  the  index  of 

relative  variation  for  efficiency  in  relation  to 

r* 

(S  —  a).     As  — s—  falls  to  zero  when  S  falls  to 

the  limiting  value  a,  and  as  it  approximates 
unity  when  S  is  indefinitely  large,  the  values 
of  the  exponent  He  between  n  and  n  —  1  for  all 
practical  cases.  In  the  second  expression  the 
coefficient  is  constant,  with  respect  to  slope, 
but  the  exponent  is  transcendental  and  intract- 
able. 

Thus  it  appears  that  the  derived  expression 
for  efficiency  as  a  function  of  S  —  a  is  not 
simply  related  to  the  coordinate  expression 
for  capacity  and  is  not  available  for  practical 
purposes;  but  it  does  not  necessarily  follow 
that  the  actual  relation  of  efficiency  to  slope 
can  not  be  formulated  for  practical  purposes  by 
an  equation  of  the  sigma  type.  All  that  is 
really  shown  is  that  if  capacity  and  efficiency 
are  both  formulated  in  that  way,  the  results 
are  not  consistent.  Formula  ( 10)  was  adopted 
for  the  capacity-slope  relation,  not  because  it 
expresses  a  demonstrated  law  of  relation,  but 
because  it  so  far  simulates  the  real  law  of  rela- 
tion as  to  be  available  for  the  marshaling  of  the 
observational  data.  It  seems  quite  possible 
that  had  the  data  been  first  translated  from 
terms  of  capacity  into  terms  of  efficiency,  the 
type  of  formula  would  have  been  found  equally 
available. 

By  way  of  testing  the  matter  a  few  com- 
parative computations  were  made,  observa- 
tional series  being  selected  for  the  purpose 
from  those  which  in  the  adjustment  gave  small 
probable  errors.  From  the  original  data  in 
Table  4  values  of  efficiency  were  computed,  and 
these  were  plotted  on  logarithmic  paper  in  rela- 
tion to  S  —  a,  the  values  of  a  being  those  em- 
ployed in  the  adjusting  equations.  In  four  of 
the  nine  cases  treated  the  locus  indicated  was  a 


KELATION  OF  CAPACITY  TO  SLOPE. 


123 


straight  line;  and  the  drawing  of  the  line  gave 
values  of  B  and  na  in  E  =  B  (S-aJ"".  In 
each  of  the  remaining  five  cases  the  indicated 
locus  was  a  curve,  and  the  curvature  was  such 
as  to  indicate  a  larger  constant  in  place  of  a. 


This  larger  constant,  a1}  was  determined  graph- 
ically, and  the  other  parameters  were  com- 
puted as  before.  The  results  are  given  in 
Table  23a,  together  with  comparative  data 
from  Table  15. 


TABLE  23a. — Comparison  of  parameters  in  the  associated  functions  of  capacity  and  efficiency,  C=6,(S— a)n  and 

E=B  (S— a,)"". 


Grade. 

w 

Q 

61 

S 

• 

»i 

»i—  » 

n 

"ll 

n-n,, 

(B) 

0.66 

0.  182  1          39.  8 

196 

0.10 

0.10 

0 

.64 

0.67 

0.97 

(B 

1.00 

.363  i          97.5 

268 

.08 

.18 

.10 

.54               .56 

.98 

f! 

.66 

.182 

38.6 

187 

.11 

.11 

0 

.54              .63 

.91 

r 

.66 

.545 

123 

193 

.06 

.06 

0 

.48              .56 

.92 

o 

i.oo 

.363 

100 

280 

.11 

.31 

.20 

.48 

.37 

1.11 

c 

1.32 

.363 

100 

240 

.16 

.22 

.06 

.40 

.54 

.86 

.66 

.734 

64.5 

78 

.36 

.56 

.20 

.69 

.74 

.91 

o 

1.00 

.734 

65.7 

63 

.41 

.41 

0 

.70 

.96 

.74 

(H) 

.66             .734 

52.9 

58.6 

.56 

.76 

.20 

.72 

.80 

.92 

Mea 

.09 

.92 

The  proximate  inferences  from  these  plots 
and  comparisons  are,  first,  that  efficiency  may 
be  formulated,  with  sufficient  accuracy  for 
practical  purposes,  as  proportional  to  a  power 
of  S  —  a1 ;  second,  that,  when  it  is  thus  formu- 
lated, the  approximate  values  of  al  are  in  gen- 
eral larger  than  the  values  of  a  obtained  in  the 
formulation  of  capacity;  and,  third,  that  the 
values  of  the  exponent  are  smaller  than  the 
equivalent  values  for  capacity,  the  differences 
usually  being  somewhat  less  than  unity. 

The  field  of  these  inferences  was  also  tra- 
versed by  a  mathematical  inquiry,  of  which 
the  results  are  more  definite.  If  the  relation 
of  efficiency  to  slope  be  formulated  by 


E=B(S-ol)n» 


-(47a) 


the  exponent  «„  is  always  less  than  n,  but  never 


so  small  as  n—  1.  For  the  range  of  conditions 
covered  by  the  experiments,  it  is  little  greater 
than  n  —  1 .  The  value  of  at  is  always  greater 
than  the  corresponding  value  of  a,  the  differ- 
ence being  usually  small.  The  difference  is 
greater  when  the  value  of  the  exponent  is 
relatively  small.  Equation  (47a)  is  incom- 
patible with  the  corresponding  equation  for 
capacity,  (10).  If  the  locus  of  E=f(S)  be 
separately  plotted  by  means  of  the  two  equa- 
tions, the  resulting  curves  are  not  coincident, 
but  they  intersect  at  three  points  and  lie  close 
together  elsewhere  (in  the  practical  field) 
unless  the  difference  between  a  and  al  is  large. 
On  the  whole,  it  appears  entirely  feasible 
to  formulate  efficiency  by  means  of  equation 
(47a). 


CHAPTER   IV.— RELATION    OF    CAPACITY  TO  FORM  RATIO. 


INTRODUCTION. 

Details  of  channel  form  in  a  natural  stream 
are  highly  diversified.  In  connection  with  the 
bendings  to  right  and  left  the  current  is  thrown 
to  one  side  and  the  other,  with  the  result  that 
the  cross  section  is  not,  for  the  most  part,  sym- 
metric about  a  medial  axis  but  shows  greater 
depth  on  the  side  of  the  swifter  flow.  In  the 
straight  channels  of  the  laboratory  there  was 
little  departure  from  bilateral  symmetry  and 
the  cross  section  was  approximately  rectangu- 
lar. For  this  reason  those  relations  of  traction 
to  form  of  cross  section  which  are  found  to  ex- 
ist in  the  laboratory  can  not,  in  general,  be  in- 
ferred of  natural  streams.  Nevertheless  there 
is  probably  an  approximate  correspondence  be- 
tween the  two  types  when  the  tractional  prop- 
erties of  a  broad,  shallow  channel  are  compared 
with  those  of  a  narrow,  deep  channel;  and  to 

that  extent  the  discussion  of  form  ratio  CR=— ) 

w' 

is  pertinent  to  the  problems  of  natural  streams. 

In  connection  with  the  study  of  the  labora- 
tory data  the  form  ratio  is  a  factor  of  great  im- 
portance, for  not  only  is  capacity  for  traction 
directly  conditioned  by  it,  but  it  affects  every 
law  of  relation  between  capacity  and  another 
condition. 

In  the  discussion  of  capacity  in  relation  to 
slope  the  effects  which  might  have  been  referred 
to  form  ratio  were  treated  instead  as  due  to 
width,  while  small  account  was  taken  of  the  co- 
ordinate influence  of  depth.  For  many  pur- 
poses the  choice  of  viewpoint  is  indifferent,  but 
when  large  and  small  channels  are  to  be  com- 
pared there  is  decided  advantage  in  taking  ac- 
count of  form  ratio.  The  form  ratios  of  labo- 
ratory channels  and  river  channels,  for  exam- 
ple, are  of  the  same  order  of  magnitude,  but  the 
widths  are  not. 

SELECTION  OF  A  FORMULA. 

MAXIMUM. 

When  identical  discharges  are  passed  through 
troughs  of  different  width  and  are  loaded  with 
d6bris  of  the  same  grade,  and  the  loads  are 
124 


adjusted  so  as  to  establish  the  same  slope,  it  is 
usually  found,  not  only  that  the  capacity  varies 
with  the  width,  but  that  some  intermediate 
width  determines  a  greater  capacity  than  do 
the  extreme  widths.  That  is,  the  curve  of 
capacity  in  relation  to  width  exhibits  a  maxi- 
mum. The  form  ratio  varies  inversely  with  the 
width;  and  the  same  maximum  appears  when 
the  capacity  is  compared  with  form  ratio.  The 
curves  in  figure  38,  introduced  to  illustrate  this 
fact,  show  data  from  Tables  12  and  14  for  grade 
(C),  with  Q= 0.363  ft.3/sec.  and  £=1.0  percent. 
In  the  upper  curve  capacities  are  compared 
with  widths;  in  the  lower  one  the  same  capaci- 
ties are  compared  with  form  ratios. 


Width 


t 

• 

u 

0 

^ 

O.I  0.2 

Form    rati  o 


FIGURE  38.— Illustration  of  the  relation  of  capacity  to  width  of  channel 
and  to  form  ratio,  when  slope  and  discharge  are  constant. 

The  formula  for  the  discussion  of  such  rela- 
tions must  be  one  affording  a  maximum.  It 
must  also  satisfy  various  physical  conditions, 
as  will  presently  appear. 

The  explanation  of  the  maximum,  so  far  as 
its  main  elements  are  concerned,  is  not  difficult. 
The  phenomenon  was  in  fact  anticipated  in  the 
planning  of  the  experiments,  and  certain 
courses  of  experimentation  were  arranged  with 
special  regard  to  the  discovery  of  the  form 
ratio  of  highest  efficiency. 

Conceive  a  stream  of  constant  discharge  and 
flowing  down  a  constant  slope  but  of  variable 
width.  The  field  of  traction  is  determined  by 
the  width,  and  the  evident  tendency  of  this 
factor  is  to  make  the  capacity  increase  as  the 
width  increases.  The  rate  of  traction  for  each 
unit  of  width  is  determined  by  the  bed  velocity 
in  that  unit,  and  the  bed  velocity  is  intimately 


BELATION   OF  CAPACITY   XO   FORM  KATIO. 


125 


associated  with  the  mean  velocity.  Velocity 
varies  directly  with  depth,  and,  inasmuch  as 
increase  of  width  causes  (in  a  stream  of  con- 
stant discharge)  decrease  of  depth,  the  tend- 
ency of  this  factor  is  to  make  capacity 
decrease  as  width  increases.  Velocity  is  also 
affected  by  lateral  resistance,  the  retarding 
influence  of  the  side  walls  of  the  channel.  The 
retardation  is  greater  as  the  wall  surface  is 
greater,  therefore  as  the  depth  is  greater,  and 
therefore  as  the  width  is  less.  As  capacity 
varies  inversely  with  the  retardation,  and  as  the 
retardation  varies  inversely  with  width,  it  fol- 
lows that  the  tendency  of  this  factor  is  to  make 
capacity  increase  as  width  increases.  Thus  the 
influence  of  width  on  capacity  is  threefold :  Its 
increase  (1)  enlarges  capacity  by  broadening 
the  field  of  traction,  (2)  reduces  capacity  by 
reducing  depth,  and  (3)  enlarges  capacity  by 
reducing  the  field  of  side-wall  resistance.  Now, 
without  inquiring  as  to  the  laws  which  affect 
the  several  factors,  it  is  evident  that  when  the 
width  is  greatly .  increased  a  condition  is  in- 
evitably reached  in  which  the  depth  is  so  small 
that  the  velocity  is  no  longer  competent  and 
capacity  is  nil.  It  is  equally  evident  that  when 
the  width  is  gradually  and  greatly  reduced 
the  field  of  traction  must  become  so  narrow 
that  the  capacity  is  very  small,  and  eventually 
the  current  must  be  so  retarded  by  side-wall 
friction  that  its  bed  velocity  is  no  longer 
competent  and  capacity  is  nil.  For  all  widths 
between  these  limits  capacity  exists,  and  some- 
where between  them  it  attains  a  maximum. 

The  forms  of  algebraic  function  which  afford 
a  maximum  are  many;  but  no  general  examina- 
tion of  them  is  necessary,  because  the  physical 
conditions  of  the  problem  serve  to  indicate  the 
appropriate  type.  As  just  observed,  the  varia- 
tion of  form  ratio  (when  discharge  and  slope  are 
constant)  involves  simultaneous  variations  of 
width  and  depth.  To  develop  an  expression 
for  the  relation  of  capacity  to  form  ratio,  it  is 
convenient  first  to  determine  separately  the 
relations  of  capacity  to  width  and  to  depth, 
and  then  to  combine  the  two  functions. 

CAPACITY   AND   WIDTH. 

To  consider  separately  the  response  of 
capacity  for  traction  to  variation  of  width  it  is 
necessary  to  relinquish,  for  the  time  being,  the 
assumption  of  constant  discharge  and  variable 
depth,  and  substitute  for  it  the  assumption  of 


constant  depth  and  slope,  with  variable  dis- 
charge. That  is,  we  are  to  conceive  a  stream 
of  constant  slope,  of  which  the  width  is  pro- 
gressively increased  or  diminished  and  of 
which  the  discharge  is  varied  in  such  way  as  to 
maintain  a  constant  depth.  Figure  39  repre- 
sents the  cross  section  of  such  a  stream,  whether 
natural  or  of  the  laboratory  type. 

Near  the  sides  the  current  is  retarded  by  side 
friction.  Also,  the  freedom  of  its  internal 
movements  is  restricted  by  the  sides,  just  as  it 
is  everywhere  restricted  by  the  upper  surface 
and  the  bed.  These  lateral  influences  diminish 
with  distance  from  the  sides  and  finally  cease  to 
be  perceptible.  We  may  thus  recognize,  in  a 
broad  stream,  two  lateral  portions,  AB,  in 
which  capacity  is  affected  by  the  sides,  and  a 
medial  portion,  AA,  in  which  capacity  is  not 
thus  affected.  In  the  medial  portion  total 
capacity  is  strictly  proportional  to  the  distance 
AA;  or,  in  other  words,  the  capacity  per  unit 


r 

S             D                             A             A. 

Z>            B; 

~~~~  ___—  "'" 

FIGURE  39.— Cross  sections  of  stream  channels;  to  illustrate  the  rela- 
tion of  capacity  to  width. 

distance,  Ct,  is  uniform.  In  a  lateral  portion 
the  capacity  per  unit  distance  diminishes  as 
the  side  is  approached.  Whatever  the  law  of 
diminution,  the  total  capacity  of  a  lateral  por- 
tion is  equivalent  to  the  capacity  per  unit  dis- 
tance in  the  medial  portion,  multiplied  by  some 
distance  AD,  less  than  AB.  Therefore  the 
total  capacity  for  the  whole  stream  is 


2DB) . 


-(48) 


C=Cl(AA  +  2AD)  =  Cl> 

It  is  evident  that  for  a  shallow  stream  the 
distances  AB  and  DB  are  less  than  for  a  deep 
stream;  and  while  the  assumption  may  not  be 
strictly  accurate,  it  must  be  approximately 
true  that  DB  is  proportional  to  the  depth. 
Making  that  assumption  and  introducing  the 
numerical  constant  «,  I  replace  2DB  by  ad, 
and  write 


As  we  are  here  concerned  only  with  the  law  of 
variation  of  C,  we  may  conveniently  replace 
this  by  the  proportion 

C*xw  —  ad. 


126 


TBANSPOBTATION   OF   DEBEIS  BY   BUNNING   WATEB. 


Substituting  for  d  its  equivalent 
Cxw(l-aR). 


d 


As  d  is  by  postulate  constant,  and  as  w  =  -^ 

H 

w  oc-g.    We  may  therefore  substitute  -=  for  w  in 
K  ti 

the  proportion  above,  obtaining 
1-aR 


Cx- 


R 


-(49) 


This  expression  gives  the  relation  of  capacity 
to  form  ratio,  so  far  as  that  relation  depends 
on  variation  of  width.  Eventually  it  is  to  be 
complemented  by  an  expression  similarly 
dependent  on  variation  of  depth. 


The  preceding  analysis  involves  the  assump- 
tion that  the  stream  is  so  broad,  in  relation  to 
its  depth,  that  its  medial  portion  is  unaffected 
by  lateral  influences.  The  resulting  proportion, 
(49),  is  not  necessarily  applicable  to  narrower 
streams.  It  is  quite  conceivable  that  when 
the  channel  is  so  narrow  that  the  reaction  of 
the  sides  affects  all  parts  of  the  current  the 
variation  of  capacity  follows  a  different  law. 
The  analytic  consideration  of  the  case  of 
narrower  channels  has  not  been  attempted, 
but  some  information  has  been  obtained  from 
the  experiments.  The  following  examination 
of  experimental  data  is  directed  toward  this 
question  and  also  toward  that  of  the  magnitude 
of  the  constant  a. 


TABLE  24. — Relation  of  capacity  for  traction  to  width  of  channel,  when  slope  and  depth  are  constant. 


Grade. 

Slope 
(per 

cent). 

Depth 

(feet). 

Value  of  C  when  width  (feet)  is— 

0.23 

0.44 

0.66 

1.00 

1.32 

l.H« 

(B) 
(C) 

(B)  and 
(C). 

0.8 

1.2 

.8 

1.2 

0.8  and 
1.2 

0.08 
.10 
.12 
.15 
.20 
.08 
.10 
.12 
.10 
.12 
.14 
.20 
.10 
.12 
.14 

.08 
.10 
.12 
.14 
.15 

8.4 
15 
21.0 
36.7 
62.5 
29 
46 
65 
12.5 
19.2 
27 
52 
45 

[i? 

[17.4] 
26.6 
38.0 
60.0 
113 
50 
74 
103 
22.0 
34 
51 
111 
80 
115 
156 

20.4 

35.4 
57.5 
112 

43 
66 
95 
145 

8.6 
15.2 
32.0 

2.8 

88 
156 

H2 
211 

290 
42 
76 
135 

22.1 
32.0 

25.0 

[5.3) 

78 

109 
151 
221 

171 

[300] 
480 

20.0 

Geometric  means. 

15.6 
29.6 
36.1 

43.1 

29.4 
50.7 
62.5 

78.2 

42.4 
75.2 

78.0 
115 

158 

211 

\. 

11.7 

124 

f 

The  assumptions  of  the  present  section 
include  constant  slope  and  constant  depth, 
with  discharge  and  capacity  adjusted  to  varia- 
tion of  width.  The  experiments  involve  con- 
stant width  and  constant  discharge,  with 
automatic  adjustment  of  slope  and  depth  to 
variation  of  load.  In  order  to  check  the 
analysis  by  means  of  the  laboratory  data  it  is 
necessary  to  employ  some  method  of  inter- 
polation. Two  methods  were  tried,  but  only 
one  need  be  described. 

Attention  being  first  restricted  to  a  par- 
ticular grade  of  d6bris  and  a  particular  slope 
of  channel,  the  computation  sheets  (p.  95) 
for  the  different  discharges  were  entered  with 
a  particular  depth  as  argument,  and  the 
associated  values  of  capacity  and  slope  were 


taken  out.  These  values  were  plotted  on 
logarithmic  section  paper  as  a  series  of  points. 
Through  these  points  was  drawn  a  curve — the 
locus  of  log  (7=/(logS),  under  the  condition 
that  d  is  constant.  By  means  of  this  curve 
values  of  C  were  interpolated,  corresponding  to 
selected  values  of  S.  The  process  was  then 
repeated  with  other  depths,  other  widths,  and 
other  grades ;  and  in  this  way  were  obtained 
sets  of  values  of  capacity  in  relation  to  width, 
under  the  condition  of  constant  depth  and 
slope.  Such  interpolated  values  of  capacity 
are  presented  in  Table  24.  It  was  found  that 
the  data  for  grades  (B)  and  (C)  only  are  full 
enough  to  serve  the  present  purpose. 

The  tabulated  capacities  are  also  plotted,  in 
relation  to  width,  in  the  upper  and  second  divi- 


RELATION    OF   CAPACITY    TO   FORM    RATIO. 


127 


sions  of  figure  40.     If  the  data  were  precise, 
ami  if  equation  (48)  were  strictly  accurate,  the 


300 


zoo 


100 


400 


300 


200 


100 


200 


100 


GracLe 


Grade 


Means 


S) 


C) 


\  2 

FIGURE  40.— Capacity  for  traction  in  relation  to  width  of  channel,  when 
depth  and  slope  are  constant.  Scale  of  capacities,  vertical;  widths, 
horizontal. 

oblique  lines  of  the  figure  would  all  be  straight 
and  would  all  intersect  the  line  of  zero  capacity 
somewhere  to  the  right  of  the  origin.  The 


irregularities  of  the  lines  are  of  such  distribution 
as  to  indicate  that  they  are  occasioned  chiefly 
by  the  imperfection  of  the  data,  and  so  far  as 
may  be  judged  by  their  inspection  the  formula 
is  substantially  correct. 

As  the  individual  lines  do  not  well  indicate 
the  points  of  intersection  with  the  horizontal 
axis,  a  set  of  composites  were  prepared,  each 
combining  the  data  for  a  particular  depth, 
without  distinction  as  to  grade  of  debris  or 
slope  of  channel.  In  the  computations  for 
these  a  few  interpolations  were  first  made,  and 
then  the  capacities  were  combined  by  taking 
their  geometric  means.  The  numerical  results 
appear  at  the  bottom  of  Table  24,  and  these 
are  represented  by  dots  in  the  lower  division  of 
figure  40.  The  indications  of  the  dots  were 
then  generalized  by  drawing  straight  lines 
among  them,  and  the  intersections  of  these 
lines  with  the  line  of  zero  capacity  gave  points 
corresponding  to  D  in  figure  39.  More  strictly, 
the  distance  of  each  intersection  from  the 
origin  gave  an  estimate  of  the  quantity  2  DB 
in  equation  (48).  As  each  estimate  is  asso- 
ciated with  a  particular  depth,  and  as  «  =  — -3 — , 

the  intersections  give  also  values  of  the  con- 
stant oi  in  (49). 


d 

2DB 

<T 

0.08 

0.30 

3.7 

.10 

.24 

2.4 

.12 

.325 

2.7 

.143 

.36 

2.5 

As  the  plotted  dots  are  so  irregular  as  to 
admit  of  much  latitude  in  the  drawing  of  the 
lines,  these  values  of  the  constant  are  far  from 
precise.  No  inference  may  be  drawn  from  their 
differences,  and  collectively  they  serve  only  to 
indicate  an  order  of  magnitude.  For  fine  sand, 
with  slopes  of  about  1  per  cent,  the  constant  « 
has  a  probable  magnitude  of  2  or  3. 

CAPACITY   AND   DEPTH. 

To  consider  separately  the  response  of  capac- 
ity to  variation  of  depth,  the  assumption  of 
constant  discharge  and  variable  width  must 
again  be  laid  aside,  and  there  must  be  substi- 
tuted for  it  the  assumption  of  constant  width 
and  slope,  with  variable  discharge.  That  is, 
we  are  to  conceive  a  stream  of  constant  width, 
of  which  the  discharge  and  load  are  simultane- 


128 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


ously  varied  in  such  way  as  to  maintain  the 
slope  unchanged,  and  the  changes  in  depth  are 
to  be  compared  with  the  changes  in  load  or 
capacity.  To  make  the  results  strictly  coordi- 
nate with  those  for  the  control  of  capacity  by 
width,  we  should  deal  only  with  channels  so 
broad  and  shallow  that  the  lateral  portions 
(fig.  39)  are  small  in  comparison  with  the  medial 
portions,  but  this  is  not  practicable.  It  is  pos- 
sible, however,  to  minimize  the  influence  of 
lateral  retardation  by  selecting  groups  of  data 
in  which  the  form  ratio  is  small. 

Table  25  contains  data  pertaining  to  debris 
of  grade  (C),  a  channel  width  of  0.66  foot,  and 
a  slope  of  1  per  cent.  The  values  of  capacity 
are  taken  from  Table  12  and  the  values  of 
depth  from  Table  14.  It  appears  by  inspection 
that  the  capacity  increases  with  the  depth. 
On  plotting  the  pairs  of  values  on  logarithmic 
section  paper,  it  is  found  that  they  may  be 
represented  approximately  by  a  straight  line. 
The  examination  of  many  such  plots  showed 
that  the  most  accurate  representative  line  has 
a  gentle  curvature,  but  for  the  present  purposes 
it  suffices  to  assume  that  the  line  is  straight. 
That  is  to  say,  it  is  found  to  be  approximately 
true,  and  the  assumption  is  made,  that  C  varies 
with  some  power  of  d,  or 


Coed" 


(50) 


In   the  particular   case   m1  =  1.90,    and   other 
values  of  m1  are  shown  in  Table  26. 

TABLE  25. —  Values  of  capacity  and  depth  for  currents  trans- 
porting debris  of  grade  (C),  when  the  width  is  0.66  foot  and 
the  slope  1 .0  per  cent. 


Discharge  (ft  '/sec  ) 

0.093 

0.182 

0.363 

0  545 

0  734 

Capacity  (gm.  /sec.)  

13.8 

32.1 

73 

112 

152 

Depth  (foot) 

.079 

.110 

.173 

228 

255 

TABLE  26. —  Valuesofml  in  Coc  rf"1',  when  slope  is  constant. 


Grade. 

Slope, 
(per 
cent). 

Value  of  mi  when  width  (feet)  Is— 

0.66 

1.00 

1.32 

1.96 

(C) 
(G) 

0.6 
1.0 

1.0 
1.6 
2.4 

2.20 
1.90 

2.67 
2.05 
1.80 

2.50 
2.10 

2.96 
3.43 
2.26 

3.10 
2.32 

3.95 
2.83 
2.48 

3.41 
2.97 

Comparison  of  the  tabulated  values  of  ml 
shows  that  they  have  considerable  range  and 


that  their  variations  in  magnitude  are  defi- 
nitely related  to  several  conditions.  The  sensi- 
tiveness of  capacity  to  variation  of  depth  (when 
width  and  slope  are  constant)  is  greater  as  the 
slope  is  less,  is  greater  as  the  fineness  is  less, 
and,  with  one  exception,  is  greater  as  the  width 
is  greater: 

m,  =/(£,  /» (51) 

As  w  is  by  postulate  constant,  and  asd—wR, 
it  follows  that  dxR.  R  may  therefore  be  sub- 
stituted for  d  in  proportion  (50),  giving 


CccR" 


.(52) 


This  is  an  expression  for  the  relation  of 
capacity  to  form  ratio,  so  far  as  that  relation 
depends  on  variation  of  depth.  It  is  the  com- 
plement of  proportion  (49)  on  page  126. 

CAPACITY   AND    FORM    RATIO. 

We  now  return  to  the  assumption  of  con- 
stant discharge.  Having  obtained  an  approxi- 
mate expression  for  the  law  of  capacity's  varia- 
tion in  response  to  change  of  width  and  a 
coordinate  expression  for  variation  on  response 
to  change  of  depth,  we  next  inquire  how  these 
may  be  combined  into  an  expression  for  the 
response  of  capacity  to  simultaneous  changes 
of  width  and  depth,  when  those  changes  are  of 
such  character  as  to  leave  discharge  constant. 

If  the  functions  (49)  and  (52)  be  assumed 
to  be  independent,  their  simple  and  direct 
combination  gives " 


C  oc 


R 


This  formula,  as  rnay  readily  be  shown,  satis- 
fies an  important  condition  by  providing  for  a 
maximum  value  of  C,  but  the  assumption  in- 
volved in  its  construction  is  not  strictly  war- 

is  based 


ranted.    The  law  embodied  in 


H 
o 


on  the  constancy  of  d  and  may  not  be  valid 
when  d  is  made  variable,  while  the  law  em- 
bodied in  R™1  may  not  be  valid  when  w  is  made 
variable. 

Viewing  the  matter  from  another  side,  we 
may  say  that  the  difficulty  would  not  exist  if 
the  values  of  oc  and  ml  depended  only  on  fine- 
ness, slope,  and  discharge  and  were  independent 
of  R;  but  they  are  in  fact  functions  of  R.  (We 
have  already  seen  that  m,  is  an  increasing 
function  of  w,  and  the  variation  of  «  with  d 


RELATION   OF   CAPACITY   TO    FORM   RATIO. 


129 


will  appear  in  another  connection.)  An  in- 
quiry as  to  the  nature  of  the  functions — an 
inquiry  here  unrecorded  except  as  to  its 
result— served  to  determine  that  the  varia- 
tions of  «  and  m^  as  functions  of  R,  affect 
capacity  in  opposite  senses,  so  that  their  in- 
fluences are  at  least  partly  compensatory.  It 
appeared '  also  that  each  influence  is  relatively 
great  when  R  is  large  and  relatively  small 
when  R  is  small,  so  that  their  laws  of  distribu- 
tion include  a  compensatory  factor. 

Despite  the  existence  of  this  difficulty, 
which  is  palliated  rather  than  cured  by  features 
of  compensation,  the  function  in  (53)  has  been 
adopted  as  the  best  practicable  formula  for 
the  relation  of  capacity  to  form  ratio.  The 
logical  defect  in  the  combination  of  its  two 
factors  may  mean  that  its  accuracy  is  inferior 
to  that  of  the  factors;  but  as  the  factors  are 
confessedly  only  approximate,  the  defect  is  not 
inconsistent  with  the  possession  of  even  superior 
accuracy  by  the  combination.  In  the  treat- 
ment of  so  intricate  a  subject  by  methods 
which  are  dominantly  empiric  altogether  ade- 
quate formulation  is  not  to  be  hoped  for;  but 
the  modicum  of  physical  foundation  afforded 
to  formula  (53)  is  believed  to  give  it  advan- 
tage over  a  purely  mathematical  expedient. 

A  slight  transformation  gives  it  more  con- 
venient form.  Moving  R  from  the  denomina- 
tor to  the  numerator  and  making  ml  —  1  =  m, 
we  have 

C  x  (1  -  a  R)  Rm 

It  is  convenient  also  to  change  from  a  propor- 
tion to  an  equation;  introducing  a  coefficient,  62, 

(7=  &2  (I  —  a  R)  Rm (54) 


<7=346  (I -2.18  R) 


.(55) 


The  corresponding  curve  is  shown  in  figure  41. 


This  conforms  to  the  physical  conditions  by 
indicating  two  values  of  R  for  which  capacity 
is  nil,  and  an  intermediate  value  for  which 
capacity  is  at  maximum.  On  referring  to  (54), 
it  is  evident  that  <7=-0  when  R  =  0,  and  also 

when  1  —  n-7?  =  0,  or  R  =  -. 

a 

In  ascribing  finite  capacity  to  all  small 
values  of  R  the  formula  is  inaccurate,  for  when 
the  form  ratio  is  gradually  reduced  the  velocity 
must  always  fall  below  competence  before  the 
ratio  reaches  zero.  The  true  function  might  be 
represented  by  some  such  curve  as  the  broken 
line  D  in  the  figure.  It  has  not  seemed  advis- 
able to  complicate  the  formula  by  a  modifica- 
tion which  might  remedy  this  defect. 

In  the  region  of  the  larger  values  of  R  the 
formula  is  subject  to  a  qualification  already 
mentioned  on  page  125.  The  larger  values  are 
associated  with  narrow  channels,  in  which  the 


As  R  is  a  ratio  between  lengths,  and  a  (p.  125) 
is  also  a  numerical  quantity  without  dimen- 
sions, &2  is  of  the  unit  of  C.  It  is  the  value  of 
capacity  when  (1  —  a  R)  Rm  —  l. 

Let  us  now  consider  the  properties  of  the 
formula.  For  the  sake  of  giving  a  visible  illus- 
tration, it  has  been  applied  to  the  example 
already  used  in  figure  38.  As  the  equation  has 
three  parameters,  its  constants  require  for  their 
determination  three  pairs  of  values  of  R  and  C. 
The  example  furnishes  four  pairs,  and  these,  by 
approximate  adjustment,  give 


FIGURE  41.—  Plot  of  equation  (55).    Capacity,  vertical;  form  ratio,  hori- 
zontal. 

influences  of  the  side  walls  or  banks  are  domi- 
nant. So  far  as  the  laboratory  data  give  indi- 
cation, the  formula  is  applicable  to  these  values; 
but  the  adjustment  might  be  less  satisfactory 
with  channel  walls  of  a  different  character. 

The  region  of  the  maximum  value  of  capac- 
ity, which  constitutes  the  chief  field  for  the 
application  of  the  formula,  is  little  affected  by 
the  qualifications  which  have  been  mentioned. 

Differentiating  equation  (54)  and  equating 
the  first  differential  coefficient  with  zero,  we 
have 


Tf\ 

whence  R  =  -       -r,  which  is  the  condition  giv- 
am+1' 

ing  C  its  maximum  value.     Designating    this 
value  of  R  by  p,  we  have 


m 


1  

nrm  +  1' 


1 

a  =  ~ 


m 


p  m  +  1 


-(57) 


20921°— No.  86—14- 


130 


TRANSPORTATION    OF   DEBRIS  BY   RUNNING   WATER. 


Substituting  from  (57)  into  (54),  we  have 


Equations  (54)  and  (58)  are  alternative  ex- 
pressions of  the  same  relation,  the  one  involving 
a  without  p,  the  other  p  without  a.  Each  has 
its  field  of  superior  convenience,  alike  for  com- 
putations and  discussion. 

To  obtain  the  maximum  capacity,  p  is  sub- 
stituted for  R  in  (58)  .     The  equation  reduces  to 


DISCUSSION  OF  EXPERIMENTAL  DATA. 

SCOPE   AND    METHOD   OF   DISCUSSION. 

The  equations  adopted  for  the  formulation  of 
C=f(R),  namely,  equations  (54)  and  (58),  in- 
volve four  constants.  &2  is  a  quantity  of  the 
unit  of  capacity;  p  is  a  ratio,  the  form  ratio  cor- 
responding to  maximum  capacity;  a  is  a  ratio 
connected  with  side-wall  resistance;  and  m  is 
an  exponent.  There  is  a  mutual  dependence 
between  a  and  p;  but  12,  a,  and  m,  grouped 
together  in  (54),  are  independent;  and  so  are 
52,  p,  and  m,  grouped  in  ( 58) .  In  the  following 
discussion  of  the  relation  of  capacity  to  form 
ratio,  equations  of  the  form  of  ( 54)  and  ( 58)  are 
derived  from  groups  of  experimental  data;  and 
these  are  compared  in  such  way  as  to  show  the 
control  of  their  constants  by  the  conditions  of 
slope,  discharge,  and  fineness. 

A  "  group  "  of  experimental  data,  for  this  pur- 
pose, includes  values  of  capacity  and  form  ratio 
from  at  least  three  observational  series,  all  per- 
taining to  the  same  fineness,  slope,  and  dis- 
charge, but  to  different  widths.  Three  pairs  of 
observational  values  suffice  to  determine  the 
three  independent  parameters;  with  a  greater 
number  the  problem  is  usually  one  of  adjust- 
ment, to  determine  the  most  probable  values  of 
the  parameters. 

The  computation  of  the  constants  bv  alge- 
braic methods  is  tedious,  and  a  graphic  method 
was  substituted.  When  an  equation  of  the 
form  of  (58)  is  plotted  on  logarithmic  section 
paper  the  shape  and  size  of  the  curve  are  deter- 
mined wholly  by  the  exponent  m:  its  position 
measured  in  the  direction  of  the  axis  of  log  R 
is  determined  by  the  value  of  p;  and  its  position 


with  respect  to  the  axis  of  log  C  by  the  value 
of  &2.  A  graphic  process  based  on  these  prop- 
erties gave  solutions  of  sufficient  approxima- 
tion for  the  purposes  of  the  discussion. 

SENSITIVENESS    AND    THE    INDEX    OF    RELATIVE 
VARIATION. 

The  sensitiveness  of  capacity  to  variation  of 
form  ratio  is  indicated  graphically  by  the  incli- 
nation of  the  logarithmic  locus  C=f(R).  As 
that  locus  is  a  curve,  the  sensitiveness  varies 
with  R.  The  form  of  the  curve,  as  mentioned 
in  the  last  paragraph,  is  determined  by  the  ex- 
ponent m,  and  the  steepness  (see  fig.  42)  of  its 
legs  varies  directly  with  m.  The  exponent  is 
thus  a  general  index  of  sensitiveness. 


FIGURE  42.— Logarithmic  plots  of  C-&2  (l Z-\  ^™, corresponding 

to  the  same  value  of  62  and  />,  but  different  values  of  m. 

The  logarithmic  equivalent  of  (54)  is 

log  (7=  log  &2  +  log  (1  -  aR)  +mlogR 
Differentiating,  we  have 

d  log  C=*d  log  (l-aR)+md  log  R 
Dividing  by  d  log  R,  we  have 

dlog  C    d  log  (1  -  aR) 
d  log  R  ~       d  log  R 

Making  substitutions  from 
dlog  C 


d  log  (1  -  aR)  •• 


-adR 
"  1  -  aR 


and 


and  reducing,  we  have 


dR 


aR 


(60) 


BELATION    OF   CAPACITY   TO    FOEM   RATIO. 


131 


It   is   evident    that 


increases    with 


increase  of  R;  therefore  i2  decreases  with  in- 
crease of  R.     Also  i2  is  positive  when  m  >  ^—  — -& 

aR 


approached    but    not    reached,    the    positive 
values  of  the  index  are  all  less  than  m.     When 

aR  =  1 ,  or  R  =  -,  the  condition  limiting  traction 


<T 


for    a    narrow,    deep    stream,    y— — „  =  +  GO 


and   negative    when    m<-j ^.     In    passing 

,.    l~.att  and    i.=  — oo.     The    negative    values    of    the 

from  positive  to   negative,  i,  passes  through  .    ,       ,,       ,                      ~    -   ., 

aft  ^                          1m  index  therefore  range  to  infinity. 

zero  when  m  =  ^ -^  or  when  R=-  ...  ,  ,  =p  The  progressive  changes  of  the  index   are 


(cf.  equation  56).      When  R  =  0,  the  limiting  conveniently  illustrated  by  the  characters  of 

aft  the  curves  in  figure  42.     Consider,  for  example, 

condition  for  a  broad,  shallow  stream,  5^^5=0  the  curve  corresponding  to  m-  1.0.     As  this 

and   i2  =  m.     As  this  condition   may  only  be  curve  is  the  logarithmic  locus  of  C=f(R),  its 


200 


Grade  (G) 
Q-.734 


S'Z.6-/, 


&6 


Grade  m 

Q.-.1S2 

\ 

S-l.8% 

\ 

1 

"""^ 

'  1.4 

1 

~'       —  l.O 

-•  — 

~o.e 

0  .Z  A  .6 

FIGUKE  43. — Relation  of  capacity,  C,  to  form  ratio,  R.    The  variation  of  the  function  C—bi(l—<zR')R'n  with  elope.     ?cale  of  C  vertical; 

scale  of  R  horizontal. 


inclination  to  the  horizontal  represents  i2.  At 
the  extreme  left,  corresponding  to  a  very  small 
value  of  R  (a  shallow,  wide  stream),  its  inclina- 
tion is  a  little  less  than  45° ;  or  the  value  of  i2  is 
a  little  less  than  1.0,  the  value  of  m.  As  the 
curve  ascends  i2  is  positive.  The  inclination 
then  diminishes  gradually  and  becomes  zero  at 
the  apex  of  the  curve,  which  corresponds  to 
R  —  p.  As  the  curve  begins  its  descent  the 
forward  inclination  corresponds  to  negative 
values  of  i2,  and  these  values  increase  numeri- 
cally, becoming  very  large  as  the  curve 
approaches  verticality. 

CONTROL   OF    CONSTANTS    BY    SLOPE. 

To  illustrate  the  way  in  which  the  index  of 
sensitiveness  and  other  constants  defining  the 
relation  of  capacity  to  form  ratio  are  influenced 


by  the  condition  slope,  two  sets  of  computa- 
tions were  made.  The  first  set  used  data  per- 
taining to  debris  of  grade  (C)  and  to  a  discharge 
of  0.182  ft.3/sec.;  the  second,  data  for  grade  (G) 
and  0.734  ft.3/sec.  The  observational  data  and 
the  computed  constants  appear  numerically  in 
Table  27  and  graphically  in  figure  43.  The 
dots  in  the  figure  represent  points  fixed  by  the 
observations,  and  through  these  have  been 
drawn  curves  conforming  to  formulas  (54)  and 
(58).  Short  vertical  lines  note  the  positions  of 
the  maxima,  where  R  =  p. 

On  referring  to  the  table,  it  is  seen  that  as  the 
slope  increases  the  values  of  m,  p,  and  a  de- 
crease, while  the  value  of  62  increases.  With 
steeper  slopes  of  channel,  capacity  is  relatively 
less  sensitive  to  changes  in  the  form  of  cross 
section;  with  steeper  slopes,  the  form  of  sec- 


132 


TRANSPORTATION    OF    DEBRIS   BY    RUNNING    WATER. 


tion  giving  highest  capacity  for  traction  is  rela- 
tively shallow  and  broad;  with  steeper  slopes, 
the  reduction  of  capacity  through  retardation 
of  the  current  by  the  channel  sides  is  relatively 
small;  and  the  general  effect  of  steepened  slopes 
is  increased  capacity. 

Each  set  of  values  of  a  constant  exhibits  an 
orderly  progression,  marred  only  by  minor  ir- 
regularities due  to  the  inaccuracies  of  graphic 

TABLE  27. — Observed  and  computed  quantities  illustrating  the 


computation.  The  results  recorded  in  Tables 
28  and  29  are  much  less  orderly.  It  is  to  be 
noted  that  the  remarkable  orderliness  in  the 
present  case  is-  not  due  to  high  precision  of 
the  data  but  to  the  fact  that  the  values  of  C 
and  R  assembled  in  Table  27  had  previously 
been  adjusted,  as  described  in  Chapter  II,  in  a 
manner  which  made  them  orderly  with  respect 
to  slope. 

influence  of  slope  on  the  relation  of  capacity  to  form,  ratio. 


Grade 
and  dis- 
charge 
(ft.'/sec.). 

Observational  data. 

Constants  of  equations. 

it 

Slope 
(per 
cent). 

Width 
(feet). 

Form 
ratio. 

Capacity 

(gm./sec.) 

m 

f 

<T 

i» 

(C) 

0.6 

0.44 

0.546 

8.7 

\ 

{—  1.55 

0.182 

.66 

.222 

12.9 

-       0.44 

0.25 

1.22 

34.4 

+  .07 

1.00 

.113 

11.4 

1 

+  .28 

1.0 

.44 
.66 
1.00 

.436 
.167 
.086 

24.7 
32.1 
31.2 

1     - 

.158 

1.05 

55.5 

f-  .65 
\  -  .01 
I  +  .10 

1.4 

.44 
.66 
1.00 

.375 
.138 
.072 

45.3 
57.0 
57.7 

)     .. 

.079 

.94 

75.2 

[-  .46 

\  -  .07 
I  +  .01 

1.8 

.66 
1.00 

.120 
.063 

87 
90 

}       [.05 

.06 

.95 

110] 

1  -  .08 
\-.01 

(G) 
0.734 

.8 

.66 
1.00 

.540 
263 

16.0 
13.3 

I       1.55 

.44 

1.38 

170 

(-1.42 
+  1.07 

1.32 

.164 

7.9    !J 

+  1.26 

1.4 

.66 

.460 

69        1 

[  -  .50 

1.00 
1.32 

.222 
.141 

65 
52 

\         .80 

.36 

1.23 

298 

{  +  .42 
I  +  .59 

2.0 

.66 

.IIS 

149 

1 

{-  .32 

1.00 
1.32 

.199 
.128 

145 
125 

1         - 

.32 

1.11 

451 

+  .27 
+  .41 

2.6 

.66 

.387 

253 

I 

|  -  .26 

1.00 
1.32 

.184 
.120 

249 
222 

1         •" 

.29 

1.07 

662 

{  +  .20 
I  +  .30 

CONTROL    OF    CONSTANTS    BY   DISCHARGE. 

Table  28  and  figure  44  exhibit  quantities  and 
curves  illustrating  the  influence  of  discharge  on 
the  relation  of  capacity  to  form  ratio.  The 
data  employed  are  those  of  grade  (C)  with  a 
channel  slope  of  1  per  cent,  and  of  grade  (G)with 
a  channel  slope  of  2  per  cent.  Of  the  seven 
curves  in  the  figure,  four  were  determined  by 
three  observational  points.  As  three  is  the 
minimum  number  of  given  points  for  the  deter- 
mination of  a  curve  of  this  type,  each  curve 
passes  through  all  the  points.  In  each  of  the 
other  cases  four  points  are  given  by  the  ob- 
servations, and  it  was  not  found  possible  to  pass 
one  of  the  curves  through  all  the  points.  If 
the  formulas  are  correct,  the  incompatibility 
indicates  errors  of  the  data.  Errors  of  similar 
magnitude  inferably  affect  the  three-point 
groups  of  data,  and  this  inference  qualifies  the 
determination  of  the  curves  and  their  equations. 
Of  the  same  general  tenor  is  the  fact  that  the 
four  curves  under  grade  (C),  pertaining  to  dis- 
charges which  increase  in  arithmetic  progres- 


sion, do  not  exhibit  an  orderly  progression  as 
to  shape. 


Grade  CO      Grade  fGJ 
S-l.0%          5-2.0%          JL 


--ZOO 


I  Q  '1.119 


.734 


100 


? 1_ 


.0  .Z  .4-  0 

FIGURE  44.— Relation  of  capacity,  C,  to  form  ratio,  R. 
of  the  function  C-iu  (1— a  R  ) 
scale  of  R  horizontal. 


.4- 

The  variation 
with  discharge.  Scale  of  ^vertical; 


RELATION   OF   CAPACITY   TO   FORM   BATIO.  133 

TABLE  28. — Observed  and  computed  quantities  illustrating  the  influence  of  discharge  on  the  relation  of  capacity  to  form  ratio. 


Grade 

Observational  data. 

Constants  of  equations. 

and 

slope 
(per 
cent). 

Dis- 
charge 
(ft.Vsec.) 

Width 
(feet). 

Form 
ratio. 

Capacity 
(gm./sec.). 

TO 

P 

a 

6, 

It 

(C) 

0.182 

0.44 

0.436 

24.7 

-0.65 

1.0 

.66 

.167 

32.1 

0.20 

0.158 

1.05 

55.5 

-  .01 

1.00 

.086 

31.2 

+  .  10 

.363 

.66 

.264 

73 

-  .84 

.00 
.32 

.130 
.082 

85 
79 

.52 

.157 

2.18 

340 

+  .13 
+  .30 

.96 

.043 

60 

+  .42 

.545 

.66 

.348 

112          | 

-  .91 

.00 
.32 

.170 
.104 

140                         M 
129          f        -40 

.174 

1.64 

392 

+  .01 
+  .19 

.96 

.054 

111 

+  .30 

.734 

.66 

.386 

152 

-  .38 

.00 
.32 

.190 
.120 

180 
187 

.05 

.061 

.78 

229 

—  .12 
-  .05 

.96 

.063 

190        i 

—  .00 

(G) 

.363 

.66 

.260 

56 

-1.15 

2.0 

.00 

.128 

45.2 

1.52 

.215 

2.80 

1,611 

+  .96 

.32 

.082 

28.3 

+  1.22 

.734 

.66 

.418 

149 

-  .32 

00 

.199 

145 

.55 

.320 

1.11 

451 

+  .27 

32 

.128 

125 

+  .41 

1.119 

66 

.532 

227 

—  .96 

00 

.260 

268 

.58 

.322 

1.14 

.830 

+  .18 

32 

.165 

238 

+  .35 

i 

The  same  incongruities  of  course  character- 
ize the  tabulated  numerical  results.  Increase 
of  discharge  is  accompanied,  in  the  main,  by 
diminution  of  ra  and  a,  but  the  indications  are 
inconclusive  as  to  p  and  62. 

CONTROL  OF  CONSTANTS  BY  FINENESS. 

With  reference  to  the  influence  of  fineness  of 
debris,  the  comparative  data  are  restricted  to 
three  grades.  Table  29  and  figure  45  record 
the  data  and  results  for  grades  (B),  (C),  and 
(G),  with  use  of  a  single  slope  and  a  single  dis- 
charge. The  curves  for  grades  (B)  and  (C), 
which  fall  close  together,  are  so  different  in 
form  as  to  intersect;  and  the  difference  shown 
by  the  forms  is  otherwise  expressed  by  incon- 
gruities among  the  values  of  constants.  With 
the  exception  of  b2,  the  constants  for  grade  (B) 
are  intermediate  between  those  of  the  other 
grades,  whereas  the  intermediate  grade  is  (C). 
If  we  ignore  this  incongruity  and  compare  the 


constants  for  grades  (B)  and  (C)  with  those  for 
the  coarse  grade  (G),  we  find  that  m,  p,  and  a 
are  all  relatively  small  for  the  fine  grades,  or 


ISO 


100 


50- 


Grade  fO> 

I 


.2  .4-  .6 

FIGURE  45.— Relation  of  capacity,  C,  to  form  ratio,  K.  The  variation 
of  the  function  C— 6s  (1— aR)  R">  withflneness  of  debris.  Scale  of  C 
vertical;  scale  of  B  horizontal. 

vary  inversely  with  fineness,  while  Z>2  varies 
directly  with  fineness.  The  variations  of  m 
and  p  are  more  pronounced  than  that  of  a. 


TABLE  29. — Observed  and  computed  quantities  illustrating  the  influence  of  the  fineness  of  debris  on  the  relation  of  capacity 

to  form  ratio. 


Observational  data. 

Constants  of  equations. 

Slope 

(per  cent) 
and  dis- 

Grade 

i, 

charge 
(ft.'/sec.). 

and 
fineness 

Width 
(feet). 

Form 
ratio. 

Capacity 
(gm./sec.). 

m 

f 

a 

!>, 

(Fi). 

0.8 

(B) 

1.00 

0.217 

138 

\ 

-0.09 

0.734 

13,400 

1.32 

.120 

139 

\      0.20 

0.16 

1.04 

242 

+  .06 

1.96 

.064 

130 

1 

+  .13 

(C) 

.66 

.440 

107 

-  .48 

5,460 

1.00 
1.32 

.214 
.134 

130 
133 

1        •" 

.126 

.85 

191 

-  .10 
-  .01 

1.96 

.070 

131 

] 

+  .06 

(G) 

.66 

.540 

16 

-1.42 

8.9 

1.00 

.263 

13.3 

(1.55 

.44 

1.38 

170 

+1.07 

1.32 

.164 

7.9 

+1.26 

134 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


SPECIAL    GROUP    OF    OBSERVATIONS. 

The  special  group  of  observations  recorded 
in  Table  4  (I),  page  51,  were  arranged  largely 
for  the  purpose  of  defining  p,  the  optimum  form 
ratio.  They  differed  from  the  main  body  of 
observations  in  that  the  interval  between  the 
discharges  employed  and  the  interval  between 
the  widths  employed  were  both  smaller;  and 
they  were  restricted  to  a  single  grade  of  debris. 
They  had  the  advantage  of  an  experimental 
method  believed  to  be  the  best  developed  in  the 
laboratory;  and  in  view  of  this  advantage  their 
series  were  constituted  of  fewer  individual  ob- 
servations than  those  of  the  main  body  of  ex- 
periments. The  results  have  not  been  satis- 
factory, and  attempts  at  formulation  in  the 
present  connection  have  developed  marked  in- 
congruities. In  figure  46  three  curves  are 
given,  and  the  corresponding  numerical  data 
appear  in  Table  30.  Each  curve  is  based  on 
five  observational  points,  but  they  are  so  irregu- 
larly placed  that  their  control  is  feeble. 

On  comparing  the  two  cases  having  the 
same  discharge,  it  is  seen  that  the  greater 
slope  is  associated  with  the  smaller  values  of 
m,  p,  and  a  and  with  the  greater  value  of  &2, 


the  difference  being  most  strongly  marked  for 
m  and  p.  On  comparing  the  two  cases  having 
the  same  slope,  it  is  seen  that  the  greater  dis- 


300- 


300 


S'l.l 
q  '1.H9 


FIGUBE  46. — Relation  of  capacity,  C,  to  form  ratio,  R.  Variation  of 
the  function  C=b?(l— aK)Em  with  slope  and  discharge.  Data  from 
special  group  of  experiments  with  debris  of  grade  (C).  Scale  of  C 
vertical;  scale  of  R  horizontal. 

charge  is  associated  with  the  smaller  values  of 
m  and  p  and  with  the  larger  value  of  62,  while 
the  values  of  «  are  nearly  equal. 


TABLE  30. — Observed  and  computed  quantities  illustrating  the  influence  of  slope  and  discharge  on  the  relation  of  capacity 

to  form  ratio. 


Observational  data. 

Constants  of  equations. 

Grade. 

Discharge 
(ft.  "/see.). 

Slope 
(per 
cent). 

Width 
(feet). 

Form 
ratio. 

C 

m 

t> 

(T 

h 

* 

l! 

(C) 

0.734 

1.1 

1.0 

0.190 

230 

1 

-0.25 

1.2 

.143 

240 

-  .07 

1.4 

.108 

240 

\        0.3 

0.123 

1.88 

586 

+  .05 

1.6 

.079 

231 

+  .12 

1.8 

.072 

230 

I 

+  .14 

1.119 

.6 

1.2 

.216 

114 

-  .72 

1.4 

.180 

125 

-  .35 

1.6 

.140 

131 

.5 

.129 

2.58 

546 

-  .06 

1.8 

.122 

125 

+  .05 

1.96 

.102 

130 

+  .15 

1.119 

1.1 

1.2 

.185 

350 

-  .34 

1.4 

.145 

400 

-  .28 

1.6 

.108 

400 

.1 

.048 

1.90 

643 

—  .16 

1.8 

.091 

418 

-  .11 

1.96 

.078 

420 

-  .07 

SUMMARY    AS    TO    CONTROL    BY    CONDITIONS. 

The  treatment  of  the  observational  data  by 
means  of  a  formula  specially  designed  to  show 
the  relation  of  capacity  for  traction  to  the 
proportions  of  the  cross  section  develops 
incongruities.  These  are  of  such  distribution 
as  to  indicate  that  they  are  due  in  chief  part 
to  the  observations  and  their  methods  of 
adjustment.  Discrepancies  which  manifestly 


pertain  to  the  data  are  so  large  that  it  is  not 
practicable  to  determine  whether  the  imper- 
fections of  the  formula  are  important. 

The  exponent  m  varies  inversely  with  slope, 
with  discharge,  and  with  fineness.  Thus  all 
the  conditions  which  tend  to  increase  capacity 
tend  also  to  make  capacity  less  sensitive  to 
changes  in  form  ratio. 

The  optimum  form  ratio,  p,  varies  inversely 
with  slope  and  with  fineness.  As  to  its  varia- 


RELATION   OF   CAPACITY   TO   FORM   RATIO. 


135 


tion  with  discharge  the  evidence  is  not  unani- 
mous; either  it  varies  inversely  under  some 
circumstances  and  directly  under  others,  or 
else  its  proper  variation  is  inverse  and  data 
of  contrary  import  are  erroneous.  The  latter 
view  is  thought  more  probable,  because  in 
many  other  connections  the  controls  of  slope 
and  discharge  follow  parallel  lines. 

The  constant  ft,  which  represents  the  resist- 
ance of  side  walls  or  banks  to  the  flow  of  the 
stream,  is  also  a  decreasing  function  of  slope, 
discharge,  and  fineness. 

The  constant  &2,  which  is  of  the  unit  of 
capacity,  varies  directly  with  slope  and  in- 
versely with  fineness,  and  the  evidence  as  to 
its  variation  with  discharge  is  conflicting.  As 
62  is  the  value  of  capacity  when 


-  aR)  R™  = 


m+1 


it  corresponds  to  a  rather  complicated  relation 
between  E,  m,  and  a,  or  R,  m,  and  p;  and  this 
relation  makes  the  interpretation  of  the  lack  of 
order  among  its  tabulated  values  a  difficult 
matter. 

If  62  is  kft  out  of  the  account,  it  is  possible 
to  generalize  by  saying  that  the  constants  of 
equations  (54)  and  (58)  vary  decreasingly  wTith 
the  conditions  which  affect  capacity  increas- 

ingly- 

(61) 

(62) 


THE  OPTIMUM  FORM  RATIO. 

The  ratio  of  depth  to  width  which  gives  to 
a  stream  its  greatest  capacity  for  traction  is  of 
importance  to  the  engineer  whenever  he  has 
occasion  to  control  the  movement  of  debris. 
The  title  optimum  ratio  is  especially  appropriate 
when  his  desire  is  to  promote  that  movement. 

The  range  of  values  for  the  ratio,  under  lab- 
oratory conditions,  is  from  1:2  to  1  :  20.  One 
effect  of  this  wide  range,  when  taken  in  connec- 
tion with  the  variety  of  conditions  by  which 
the  ratio  is  controlled,  is  to  complicate  the 
formulation  of  practical  rules;  but  this  diffi- 
culty is  not  insuperable.  It  is  qualified  to  an 
important  degree  by  the  consideration  that 
capacity,  in  the  region  of  its  maximum,  changes 
very  slowly  with  change  of  form  ratio,  so  that 


an  approximate  determination  of  the  ratio  has 
practical  value. 

The  values  of  the  ratio  given  in  Table  31  are 
appropriate  to  the  conditions  of  the  Berkeley 
laboratory — that  is,  they  pertain  to  troughs  a 
few  inches  or  a  few  feet  wide,  with  smooth 
vertical  sides.  It  is  important  to  note  also  that 
they  apply  only  to  transportation  of  debris 
over  a  bed  of  debris,  and  not  to  flume  traction, 
which  has  a  different  law.  (See  p.  213.) 

TABLE  31. — Estimated  ratios  of  depth  of  current  to  width  of 
trough,  to  enable  a  given  discharge,  on  a  given  slope,  to 
transport  its  maximum  load. 


Material  transported. 

Slope 
(per 
cent). 

Ratio  for  discharge  (ft.8/sec.)  of  — 

0.25 

0.50 

0.75 

1.00 

[      0.5 
]      1.0 
I     2.0 

(      1.0 
\      2.0 

\      3.0 

1:4 
1:6 
1:10 

1:6 
1:9 
1:16 

1:8 
1:12 
1:20 

1:25 
1:3 
1:4 

1:9 
1:15 

Coarse  sand  or  fine  gravel  

1:3 
1:5 

1:7 

""i:2" 

1:25 
1:3 

No  way  has  been  found  to  extend  the  quan- 
titative results  to  rivers.  It  can  hardly  be 
questioned  that  the  optimum  ratio  for  rivers 
varies  inversely  with  slope,  discharge,  and 
fineness  of  debris,  but  its  absolute  amount  can 
not  be  inferred  from  the  experimental  results. 
River  slopes  are  relatively  very  small  and  river 
discharges  are  relatively  very  large,  and  the 
two  differences  affect  the  ratio  in  opposite 
ways.  To  compute  the  joint  result  we  should 
have  definite  and  precise  information  as  to  the 
laws  of  dependence,  but  our  actual  knowledge 
is  qualitative  and  vague. 

In  this  connection  it  is  of  interest  to  record  a 
single  observation  on  river  efficiency.  Where 
Yuba  river  passes  from  the  Sierra  Nevada  to 
the  broad  Sacramento  Valley  its  habit  is  rather 
abruptly  changed.  In  the  Narrows  it  is  nar- 
row and  deep;  a  few  miles  downstream  it  has 
become  wide  and  shallow.  Its  bed  is  of 
gravel,  with  slopes  regulated  by  the  river 
itself  when  in  flood,  and  the  same  material 
composes  the  load  it  carries. 

In  the  Narrows  the  form  ratio  during  high 
flood  is  0.06  and  the  slope  is  0.10  per  cent. 
Two  miles  downstream  the  form  ratio  is  0.008 
and  the  slope  is  0.34  per  cent.  Thus  the 
energy  necessary  to  transport  the  load  where 
the  form  ratio  is  0.008  is  more  than  three  times 
that  which  suffices  where  the  form  ratio  is 


136 


TRANSPORTATION    OF    DEBBIS   BY    RUNNING    WATER. 


0.06;  and  it  is  evident  that  the  larger  ratio  is 
much  more  efficient  than  the  smaller.  The 
data  do  not  serve  to  define  the  optimum  form 
ratio,  but  merely  show  that  it  is  much  greater 
than  0.008.  In  this  instance  the  slopes  and 
fineness  are  of  the  same  order  of  magnitude  as 
those  realized  in  the  laboratory,  but  the  dis- 
charges are  of  a  higher  order. 

When  water  without  detrital  load  is  con- 
veyed by  an  open  rectangular  conduit,  the 
form  ratio  of  highest  efficiency  is  that  which 
yields  the  highest  mean  velocity.  It  is  ap- 
proximately 1:2.  This  corresponds  to  the 
maximum  value  of  the  optimum  ratio  for  trac- 
tion, and  the  correspondence  might  have  been 
expected  on  theoretic  grounds.  The  two  fac- 
tors which,  in  ultimate  analysis,  determine 
capacity  for  traction  are  velocity  of  current 
along  the  bed  and  width  of  bed.  When  dis- 
charge and  slope  are  such  as  barely  to  afford 
competence  with  the  most  favorable  form 
ratio,  that  ratio  is  one  giving  the  highest 
velocity,  namely,  1:2.  The  other  factor, 
width  of  bed,  is  evidently  favored  by  lower 
values  of  R;  and  therefore,  as  the  conditions 
recede  from  the  limit  of  competence,  the  opti- 
mum form  ratio  becomes  smaller.  This  line 
of  reasoning  might,  in  fact,  have  been  used  to 
show  a  priori — what  has  actually  been  shown 
by  the  experiments — that  the  value  of  p  varies 
inversely  with  slope,  discharge,  and  fineness. 

SUMMARY. 

Capacity  for  traction  varies  with  the  depth 
of  the  current,  being  approximately  (though 
not  precisely)  proportional  to  a  power  of  the 
depth.  Capacity  varies  also  with  the  width 
of  the  current,  being  approximately  propor- 


tional to  the  width  less  a  constant  width. 
This  constant  width  is  equivalent  to  the  prod- 
uct of  the  depth  by  a  numerical  constant. 
When  the  discharge  is  constant,  any  change  of 
width  causes  a  change  of  depth  and  also  a 

change  of  form  ratio,  R  =  -.     A  formula  for  the 

variation  of  capacity  in  relation  to  form  ratio, 
when  the  discharge  is  constant,  is  based  on  the 
above-mentioned  properties  and  takes  the 
form 

C=l2(l-aR)Rm  ..(54) 

in  which  a  is  a  numerical  constant;  or 


in  which  p  is  the  optimum  form  ratio,  or  the 
form  ratio  giving  the  highest  capacity. 

That  capacity  should  have  a  maximum  value 
corresponding  to  some  particular  value  of  form 
ratio  is  made  to  appear  from  theoretic  con- 
siderations, and  the  fact  of  a  maximum  is 
shown  by  the  experimental  data.  The  same 
data  show  that  the  optimum  form  ratio  has 
different  values  under  different  conditions,  its 
values  becoming  smaller  as  slope,  or  discharge, 
or  fineness  increases. 

The  sensitiveness  of  capacity  to  the  control 
of  form  ratio  is  indicated  in  the  formulas  by 
the  exponent  of  R,  and  that  also  varies  with 
conditions.  It  becomes  smaller  as  slope,  or 
discharge,  or  fineness  increases. 

It  is  believed  that  all  the  generalizations 
from  the  laboratory  results  may  be  applied  to 
natural  streams,  but  only  in  a  qualitative  way; 
the  disparity  of  conditions  is  so  great  that  the 
numerical  results  can  not  be  thus  applied. 


CHAPTER  V.— RELATION  OF  CAPACITY  TO  DISCHARGE. 


FORMULATION  AND  REDUCTION. 

As  a  condition  controlling  the  capacity  of  a 
stream  for  the  traction  of  debris,  discharge  is 
the  coordinate  of  slope.  Each  of  the  two  fac- 
tors is  proportional  to  the  potential  energy  of 
the  stream,  on  which  traction  depends;  and  the 
control  of  each  is  exercised  through  the  control 
of  velocity.  Their  fundamental  difference  in 
relation  to  traction  is  connected  with  depth  of 
current.  When  velocity  is  augmented  by  in- 
crease of  slope,  the  depth  is  reduced;  when  it  is 
augmented  by  increase  of  discharge,  the  depth 
is  increased.  Notwithstanding  this  difference, 
the  relations  of  capacity  to  discharge  parallel 
those  of  capacity  to  slope  to  a  remarkable 
extent.  Thanks  to  this  parallelism,  the  discus- 
sions of  the  present  chapter  may  be  based  in 
considerable  part  on  those  which  have  preceded. 

The  data  for  the  comparison  of  capacity  with 


discharge  are  contained  in  Table  12.  In  each 
division  of  that  table  assigned  to  a  grade  of 
d6bris  are  a  number  of  subdivisions  pertaining 
severally  to  particular  widths  of  channel.  Each 
column  of  such  a  subdivision  pertains  to  a  par- 
ticular discharge  and  contains  a  series  of  ad- 
justed capacities  corresponding  to  an  orderly 
series  of  slopes.  The  values  of  capacity  con- 
nected with  the  same  slope  and  comprised  in 
the  same  subdivision  constitute  a  group  illus- 
trating the  relation  of  capacity  to  discharge. 
Table  32  contains  in  its  upper  part  a  number  of 
such  groups,  selected  and  arranged  for  the  pres- 
ent purpose.  So  far  as  practicable  they  per- 
tain to  the  same  slope,  but  it  was  not  possible  to 
secure  absolute  uniformity  in  this  respect  and 
at  the  same  time  make  the  representation  in- 
clude data  of  all  the  grades  of  debris.  For 
grades  (A)  to  (E)  the  slope  is  1.0  per  cent;  for 
grades  (F)  to  (H),  1.2  per  cent. 


TABLE  32. — Data  on  the  relation  of  capacity  to  discharge,  with  readjusted  values  of  capacity,  Cr,  and  values  of  the  index  of 

relative  variation,  ia. 


{Grade              

(A) 
1.0 
.66 

(A) 
1.0 
1.00 

(A) 
1.0 
1.32 

(A) 
1.0 
1.96 

(B) 
1.0 
.23 

(B) 
1.0 
.44 

(B) 
1.0 
.66 

(B) 
1.0 
1.00 

(B) 
1.0 
1.32 

(B) 
1.0 
1.96 

Slope  (p 

sr  cent)  .  . 
reel)  

Width  ( 

Data 

Q 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

0.093 
.182 
.363 
.545 
.734 
1.119 

[14.8] 
39.5 

7.8 
13.3 

13.2 

28.7 

10.8 
33.5 
81 
120 

37.5 
100 

36.8 
104 

30.1 
85 
143 
199 

29.3 
79 
140 
204 

96 

67.6 
120 
190 
313 

140 

231 

250 

240 
359 

(       "• 
I      ft, 

.028 
1.08 
290 

.048 
1.05 
370 

.067 
1.09 
389 

.105 
1.00 
367 

.002 
.79 
49.5 

.017 
1.00 
170 

.033 
1.13 
270 

.057 
1.12 
313 

.080 
1.06 
313 

.125 
1.08 
312 

Readjusted  capacities  

« 

Cr 

Cr 

Cr 

Cr 

Cr 

Cr 

Cr 

Cr 

Cr 

Cr 

0.093 
.182 
.363 
.545 
.734 
1.119 

15.1 
38.5 
89 
142 

7.8 
13.3 

13.2 
28.7 

11.1 
31.6 

77 
127 

37.5 
99 
167 
231 

37.0 
103 
174 
250 

30.3 
83 
140 
201 

28.5 
83 
140 
201 

96 
159 
234 
370 

67.6 
123 
185 
311 

Index  of  relative  variation  

Q 

fa 

fa 

fa 

fa 

fa 

is 

k 

>, 

k 

i> 

0.093 
.182 
.363 
.545 
.734 
1.119 

1.54 

1.28 
1.17 
1.14 

0.81 
.80 

1.21 
1.10 

1.75 
1.38 
1.24 
1.20 

1.43 
1.21 
1.15 
1.12 

1.72 
1.33 
1.24 
1.20 

1.63 
1.33 
1.25 
1.21 

1.89 
1.36 
1.24 
1.19 

1.41 
1.24 
1.17 
1.10 

1.64 
1.40 
1.30 
1.21 

Probable  error  (per  cent)  

/        C 
\        Cr 

4.8 
2.4 

1.4 

0.7 

2.4 
1.2 

1.4 
0.7 

137 


138 


TRANSPORTATION   OP   DEBKIS  BY   RUNNING   WATER. 


TABLE  32. — Data  on  the  relation  of  capacity  to  discharge,  with  readjusted  values  of  capacity,  CT,  and  values  of  the  index  of 

relative  variation,  % — Continued. 


{Grade    

(C) 
1.0 

.44 

(C) 
1.0 

.66 

(C) 
1.0 
1.00 

(C) 
1.0 
1.32 

(C) 
1.0 
1.96 

CO) 

1.0 

.66 

(D) 
1.0 
1.00 

(D) 
1.0 
1.32 

(E) 
1.0 

.66 

Slope  (per  cent).. 
Width  (feet)  

Data  

Q 

C 

C 

c 

C 

C 

C 

c 

C 

C 

0.093 
.182 
.363 
.545 
.734 
1.119 

10.9 
24.7 

13.8 
32.1 
73 
112 
152 

9.1 

31.2 
85 
140 
180 
276 

21.4 
79 
129 
187 

29.8 

24.0 
73 
108 
153 

60 
111 
190 
343 

59.2 

24.8 

101 

131 

40 

1        '• 
\       »J 

.020 
1.02 

157 

.041 
.94 
213 

.071 

1.00 
285 

.099 
1.05 
305 

.156 
1.14 
354 

.059 
.90 

195 

.102 
.88 
224 

.143 
.81 
201 

.108 

Q 

Cr 

Cr 

Cr 

C, 

Cr 

Cr 

Cr 

Cr 

Cr 

0.093 

.182 
.363 
.545 
.734 
1.119 

10.9 

24.7 

13.2 
33.9 
74 
112 
151 

9.2 
29.6 
67 
102 

33.0 
84 
135 

187 

m 

22.0 
75 
130 
190 

24.3 
69 
110 
151 

59 
120 
190 
340 

59.2 
97 
131 

Index  of  relative  variation  

Q 

's 

(| 

k 

k 

is 

k 

d 

is 

is 

0.093 

.182 
.363 
.545 
.734 
1.119 

1.30 
1.14 

1.68 
1.21 
1.06 
1.02 
1.00 

2.46 

1.64 
1.24 
1.15 
1.11 

1  07 

2.30 
1.44 
1.28 
1.21 

1.33 
1.07 
1.01 

2.00 
1.22 
1.08 
1.02 

2.00 
1.60 
1.45 
1  32 

1.33 
1.10 
1.01 

/        C 

\     c. 

2.2 
1.0 

3.6 
1.6 

2.7 
1.3 

2.9 
1.5 

2.4 

Probable  error  (per  cent)  

1.2 

(Grade  

(E) 
1.0 
1.00 

(E) 
1.0 
1.32 

(F) 
1.2 
.66 

(F) 
1.2 
1.00 

(F) 
1.2 
1.32 

(G) 
1.2 

.66 

(0) 
1.2 
1.00 

(G) 
1.2 
1.32 

(II) 
1.2 

.66 

Slope  ( 

percent)., 
(feet)  

Width 

Eata  

« 

c 

C 

C 

C 

C 

C 

C 

C 

C 

0.093 
.182 
.363 
.545 
.734 
1.119 

14.8 
33.8 

9.3 
29.9 

4.2 
22.8 

36.3 

21.4 

14.8 

9.7 

3.1 

72 
138 

73 
123 

55.3 

63 
112 

61.6 
101 

48.3 
75 

44.6 
92 

33.8 
81 

24.6 
43.5 

!  • 
i  (>s 

.188 

.263 

.139 

.242 

.339 

.199 
.94 

.345 

.433 

.233 
1.24 

.54 

.83 

Q 

Cr 

Cr 

Cr 

Cr 

Cr 

Cr 

Cr 

Cr 

Cr 

0.093 
.182 
.363 
.545 
.734 
1.119 

15.1 

4.2 

46 

76 

22.5 
46 

Index  of  relative  variation  

<3 

is 

<3 

is 

is 

is 

ll 

k 

fa 

l| 

0.093 

.182 
.363 
.545 
.734 
1.119 

1 

2.04 

3.46 

1  29 

1.82 
1.57 

1  22 

Probable  error  (per  cent)  

1        C 
1         Cr 

BELATION   OF   CAPACITY   TO   DISCHARGE. 


139 


The  same  data  are  plotted  in  figure  47,  where 
horizontal  distances  represent  logarithms  of  dis- 
charge and  vertical  distances  logarithms  of  ca- 
pacity. For  each  of  the  above-mentioned 
"  groups  "  the  plotted  points  are  connected  by  a 
series  of  straight  lines,  and  each  of  the  broken 


FIGURE  47. — Logarithmic  plots  of  the  relation  of  capacity  to  discharge. 
The  horizontal  scale  is  that  of  log  Q,  the  vertical  of  log  C.  The  zeros 
of  log  C  for  the  different  plots  are  not  the  same. 

lines  thus  produced  is  the  rough  logarithmic 
graph  of  an  equation  C=*f(Q).  The  graphs 
are  arranged  according  to  grades  of  debris,  and 
secondarily  according  to  widths  of  channel, 
their  order  from  left  to  right  corresponding  to 
the  sequence  from  narrower  to  broader  channels. 
In  effecting  this  arrangement  graphs  were 


moved  bodily  up  or  down  but  not  to  the  right  or 
left. 

It  appears  by  inspection  that  the  graphs 
bend  toward  the  right  as  they  ascend.  To  this 
rule  there  are  a  few  exceptions,  but  the  only 
strongly  marked  exceptions  are  connected  with 
grade  (E),  the  data  for  which  have  previously 
been  recognized  as  anomalous.  As  the  incli- 
nation of  each  line  indicates  the  sensitiveness  of 
capacity  to  the  control  of  discharge,  the  general 
bending  to  the  right,  or  the  reduction  of  inclina- 
tion in  passing  from  lower  to  higher  discharges 
shows  that  sensitiveness  diminishes  as  dis- 
charge increases.  This  feature  is  similar  to  one 
observed  in  studying  the  relation  of  capacity  to 
slope,  and  as  that  feature  was  found  to  be  con- 
nected with  competence,  the  resemblance  leads 
at  once  to  the  suggestion  that  here  also  is  a  con- 
nection with  competence. 

If  a  very  small  discharge  be  made  to  flow 
over  a  sloping  bed  of  debris  and  the  discharge 
be  gradually  increased,  transportation  of  ddbris 
wiU  commence  when  the  competent  discharge 
is  reached  and  will  increase  with  further  in- 
crease of  discharge.  It  is  a  plausible  hypothe- 
sis that  the  capacity  is  more  simply  related 
to  the  excess  of  discharge  above  the  competent 
quantity  than  to  the  total  discharge.  Follow- 
ing the  procedure  in  the  case  of  capacity  and 
slope  we  may  assume  that  capacity  is  propor- 
tional to  a  power  of  the  excess  of  discharge 
above  a  constant  discharge,  the  constant  dis- 
charge being  closely  related  to  competent  dis- 
charge. In  the  following  formula,  constructed 
on  the  plan  of  equation  (10), 

C=13(Q-K}° ...(64) 

K  is  a  constant  discharge  and  &3  is  a  constant 
numerically  equal  to  the  value  of  capacity 
when  the  discharge  equals  x+1,  although  it  is 
not  strictly  a  capacity.  The  dimensions  of 
capacity  are  M+l  T'1,  of  discharge  L+3  T'1,  and 
of  (Q-K)°  L+3°  T-°;  and  these  values  give  to 
63  the  dimensions  L'30  M+l  T°~l. 

As  a  preliminary  to  the  adjustment  of  the 
observational  data  of  Table  32  by  this  formula, 
values  of  «  were  graphically  computed  by  the 
method  previously  employed  in  connection 
with  a.  The  computations  were  applied  to  all 
groups  of  values  of  capacity  in  the  table, 
except  such  as  comprise  less  than  three  capaci- 
ties and  except  also  the  aberrant  data  of  grade 
(E).  In  Table  33  the  results  are  arranged 


140 


THANSPORTATION    OF    DEBRIS    BY    RUNNING    WATER. 


with  reference  to  grade  of  debris  and  channel 
width. 

TABLE  33. —  Values  of  the  discharge  constant,  K,  computed 
from  the  discharges  and  capacities  of  Table  32. 


Grade. 

Value  of  K  when  width  of  channel 
(in  feet)  is  — 

0.66 

1.00 

1.32 

1.9C 

i 

C 

I,' 

ii 

0.070 
.095 
.100 
.130 
.130 
.160 

0.075 
.035 
.030 

0.215 
.000 
.060 

0.056 
.025 
.060 
.149 
.280 
.326 

.180 

The  tabulated  values  of  K  show  great  irreg- 
ularity. It  is  not  difficult,  however,  to  recog- 
nize a  tendency  to  grow  larger  in  passing  from 
finer  debris  to  coarser;  and  there  is  also, 
though  it  is  ill  defined,  a  tendency  to  increase 
with  increasing  width.  The  cause  of  the  first- 
mentioned  tendency  is  readily  understood, 
because  relatively  coarse  debris  requires  a 
relatively  swift  current  to  move  it;  and  the 
reality  of  the  second  also  finds  support  when 
the  conditions  affecting  competent  discharge 
are  considered. 

Postulating,  initially,  a  channel  of  some 
particular  width,  containing  a  stream  of  which 
the  discharge  is  barely  competent,  let  us  assume 
that  the  width  is  increased.  In  spreading  to 
the  new  width  the  stream  loses  depth  and 
velocity,  and  its  velocity  is  no  longer  compe- 
tent. That  the  velocity  may  again  become 
competent  the  discharge  must  be  increased. 
Thus  it  is  in  general  true  that  competent  dis- 
charge increases  with  increase  of  width.  In 
the  cross  section  of  a  broad  laboratory  current, 
figure  48,  a  medial  portion,  AA,  is  unaffected 


FIGURE  48.—  Ideal  cross  section  of  a  stream  in  the  experiment  trough, 
illustrating  the  relation  of  competent  discharge  to  width. 

by  side-wall  resistance  and  has  competent 
velocity.  Two  lateral  portions,  AB,  have  less 
than  competent  velocity.  Let  us  imagine 
these  lateral  portions  replaced  by  narrower 
divisions,  AD,  in  which  velocities  are  the  same 
as  in  AA.  The  effective  width  for  the  main- 
tenance of  competent  velocity  is  then  DD  = 


w  —  2  BD.  If  now  width  and  discharge  be 
increased  or  diminished,  with  maintenance  of 
competent  velocity,  the  quantity  2  BD  is 
unaffected,  so  that  it  may  be  regarded  as  a 
constant.  Velocities  being,  by  hypothesis, 
uniform  through  the  whole  space  DD,  the  dis- 
charge is  proportional  to  the  width  of  that 
space. 

Qc  <x  w  —  &  constant (65) 

This  result  assumes  a  channel  so  wide  that  a 
medial  portion  is  unaffected  by  side-wall  resist- 
ance. It  may  not  be  true  for  narrower  chan- 
nels. If  it  were  to  be  refined,  the  constant 
would  be  found  to  be  a  function  of  depth  (com- 
pare p.  129);  but  in  applying  its  principle  to 
the  values  of  «  no  aUowance  was  made  for 
that  factor. 

On  the  theory  that  K  is  closely  related  to 
competent  discharge  and  has  similar  properties, 
and  with  the  assumption  that  (65)  may  be 
applied  to  narrow  channels  as  weU  as  broad,  the 
principle  of  (65)  was  used  in  adjusting  the 
tabulated  values  of  K  in  relation  to  width  of 
channel.  The  formula  assumed  was 

K  <x  w  — 0.2 

The  value  of  the  constant  was  arbitrarily 
fixed,  after  several  trials,  no  criterion  for  selec- 
tion being  discovered  except  the  harmony  of 
results. 

With  the  aid  of  this  formula  it  was  possible 
to  combine  the  data  of  Table  33  in  such  way  as 
to  afford  a  better  view  of  the  relation  of  K  to 
grade,  or  fineness ;  and  this  was  done.  By  mul- 
tiplication or  division,  each  value  was  reduced 
to  its  equivalent  for  a  channel  width  of  1  foot, 
and  means  were  taken.  These  means  appear 
in  the  second  line  of  Table  34.  They  were  com- 
pared with  the  mean  diameters  of  particles  for 
the  several  grades,  the  comparison  being  made 
on  logarithmic  section  paper.  In  figure  49, 
showing  this  plot,  the  numbers  indicate 
weights.  While  the  plotted  points  do  not  fall 
well  into  line,  they  leave  no  question  that  the 
value  of  K  rises  as  the  debris  becomes  coarser. 
On  the  assumption  that  the  function  is  a  power 
function,  a  straight  line  was  drawn  to  represent 
it;  and  by  this  line  the  mean  values  of  K  were 
adjusted.  Values  for  the  other  trough  widths 
were  then  computed.  These  appear  in  Table  32. 


BELATION   OF   CAPACITY   TO   DISCHARGE.  141 

TABLE  34. — -Adjustment  of  system  of  values  of  K  for  a  channel  width  of  1  foot  and  slopes  of  1.0  and  1.2  per  cent. 


Grade        

(A) 

(B) 

(C) 

(D) 

(E) 

(F) 

(O) 

(H) 

0.00100 

0.  00123 

0.00166 

0.  00258 

0  00561 

0  01040 

0.  01620 

0.02300 

.074 

.078 

.047 

.117 

.188 

.367 

.925 

.048 

.057 

.071 

.102 

.188 

.242 

.345 

.406 

In  making  these  adjustments  due  account 
was  taken  of  the  fact  that  not  all  the  data  used 
pertain  to  the  same  slope.  Allowance  for  slope 
difference  was  made  in  plotting  the  values  of 
K  to  adjust  for  fineness,  and  again  in  applying 
the  results  of  that  adjustment.  The  adjust- 
ments for  slope  are  based  on  the  following 
consideration. 

By  use  of  the  data  in  Table  12,  values  of  K 
pertaining  to  the  same  grade  and  width  but 
differing  as  to  slope  were  compared.  In  one 
case  K  appeared  to  increase  with  increase  of 
slope,  but  in  most  cases  it  appeared  to  decrease. 


FIGPBE  49.— Logarithmic  plot  otn-f(D'). 

Little  weight,  however,  was  given  to  such 
determinations,  because  it  was  evident  that 
they  were  controlled  by  the  values  of  a,  which 
had  been  somewhat  arbitrarily  assigned  in  the 
first  adjustment  of  the  observations.  A  more 
satisfactory  indication  was  obtained  by  reason- 
ing based  on  the  relative  sensitiveness  of 
capacity  to  the  several  controls  of  slope  and 
discharge.  The  index  of  relative  variation  of 
capacity  with  respect  to  slope  being  i1;  we  may 
designate  the  corresponding  index  with  respect 
to  discharge  by  is.  As  capacity  varies  simul- 
taneously with  S'1  and  Q'3  it  is  evident  that  for 
any  particular  value  of  capacity  5"1  and  Q'' 


vary  inversely, 
yields 


The    proportion 


S7' 


oc 


From  a  discussion  of  the  relations  of  the  two 
indexes,  to  be  found  on  a  later  page  of  the 
present  chapter,  it  appears  that  the  average 
ratio  of  i,  to  i3  is  1.36  but  that  the  ratio  varies 
somewhat  with  conditions.  For  the  condition 

1 


£=0  it  is  assumed  to  be  1.3;  or  K  oc 


S1-3 


While  considerable  uncertainty  attaches  to 
the  numerical  relations,  there  can  be  no  ques- 
tion that  K  varies  directly  with  width  and 
inversely  with  fineness  and  slope.  As  any 
change  in  width  affects  the  form  ratio  in  the 
opposite  sense,  we  may  write 

,fy  ........  -.-(66) 


A  system  of  values  of  «  having  been  thus 
arranged,  logarithmic  plots  were  made  of  the 
capacities  of  Table  32,  in  relation  to  Q  —  K,  and 
straight  lines  were  drawn  representing  equa- 
tions of  the  type  of  (64).  In  cases  where  the 
plotted  points  did  not  fall  well  in  line,  recourse 
was  had  to  weights  based  on  the  probable 
errors  recorded  in  Table  12.  The  lines  thus 
drawn  gave,  by  their  inclinations  and  intersec- 
tions, the  values  of  the  other  parameters,  o  and 
63,  as  well  as  the  adjusted  values  of  capacity, 
CT,  all  of  which  are  presented  in  Table  32.  The 
plots  for  data  of  grades  (E)  and  (F)  and  for 
two  of  the  three  groups  under  grade  (G) 
exhibited  so  great  irregularities  that  it  was 
decided  not  to  use  them. 

The  similarity  of  equation  (64)  to  equation 
(  10)  makes  it  unnecessary  to  repeat  the  reason- 
ing by  which  (38)  was  educed,  and  the  equation 
for  the  index  of  relative  variation  for  capacity 
in  relation  to  discharge  may  be  written  directly: 


I** 


3  ~  Q  —  i 


(67) 


By  its  use  the  values  of  i3  in  Table  32  were 
computed. 


142 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


MEASURES    OF    PRECISION    AND    THEIR 
INTERPRETATION. 

In  16  of  the  columns  of  Table  32  the  values 
of  capacity  before  and  after  adjustment  consti- 
tute data  suitable  for  the  computation  of 
probable  errors.  As  the  number  of  residuals 
is  in  each  case  small,  the  particukr  values  of 
probable  errors  (recorded  at  the  bottom  of  the 
table)  are  not  themselves  of  high  precision;  but 
the  averages  have  greater  claim  to  attention. 
It  should  be  added  that  the  residuals  were 
treated  as  of  equal  weight,  despite  the  fact 
that  in  the  adjustment  which  determined  them 
the  relative  weights  of  the  observational  data 
were  recognized. 

The  probable  errors  of  the  upper  line,  marked 
"  C,"  pertain  to  the  adjusted  values  of  capacity 
taken  from  Table  12,  which  are  here  treated  as 
observations.  The  probable  errors  of  the  lower 
line,  marked  "  Cr,"  pertain  to  the  groups  of 
readjusted  values  of  capacity. 

The  average  probable  error  (or,  strictly,  the 
mean  of  the  nine  tabulated  errors)  of  the  read- 
justed capacities  is  ±1.3  per  cent.  The  cor- 
responding average  for  the  capacities  before 
readjustment  is  ±2.64  per  cent. 

It  is  of  interest  to  compare  the  last  figure 
with  the  previously  computed  average  probable 
error  of  the  adjusted  values  of  capacity  in 
Table  12,  namely,  ±2.50  per  cent.  The  earlier 
estimate  applies  to  66  series  of  values,  the  later 
to  36  values  taken  from  36  of  the  series.  The 
36  values  may  be  regarded  as  properly  repre- 
sentative of  the  series  from  which  they  come. 
Of  the  unrepresented  series,  a  portion  escaped 
because  the  values  they  furnished  to  Table  32 
fell  in  groups  of  less  than  four,  and  such  a 
group  did  not  afford  suitable  data  for  computa- 
tion of  probable  error.  The  others  were  omitted 
because  the  groups  they  constituted  involved 
incongruities  so  great  that  they  were  rejected  in 
the  readjustment.  The  omission  of  incongru- 
ous groups  evidently  had  the  effect  of  lowering 
the  estimate  of  average  probable  error,  and  to 
give  validity  to  the  comparison  due  allowance 
should  be  made  for  that  effect.  The  discarded 
values  were  accordingly  treated  so  far  as  neces- 
sary to  compute  their  probable  errors,  and  it  was 
found  that  by  the  inclusion  of  these  the  estimate 
of  average  probable  error  was  raised  from  2.64  to 
3.2  per  cent.  The  revised  estimate  is  believed 


to  be  properly  comparable  with  the  earlier  esti- 
mate of  2.50  per  cent.  It  will  be  recalled  that 
the  original  observations  were  characterized  by 
accidental  errors,  ascribed  chiefly  to  rhythm,  and 
by  systematic  errors,  ascribed  chiefly  to  methods 
of  observation.  The  nature  of  the  first  adjust- 
ment was  such  that  its  computed  probable 
errors  were  little  affected  by  the  errors  of  the 
second  class.  The  adjustment  was  condi- 
tioned by  a  formula  involving  the  assumption 
that  capacity  varies  as  a  power  of  (S-a),  and 
also  by  various  assumptions  involved  in  the 
arrangement  of  a  system  of  values  for  a.  What- 
ever errors  were  introduced  in  connection  with 
these  assumptions  tended  to  increase  the  esti- 
mates of  probable  error;  but  they  may  also  be 
supposed  to  have  aggravated  somewhat  the 
errors  of  the  class  not  covered  by  the  computa- 
tions of  probable  error. 

The  errors  falling  outside  the  estimates  of 
probable  error  were  largely  of  such  nature  as  to 
affect  an  observational  series  in  its  entirety, 
and  it  was  expected  that  they  would  be  re- 
vealed in  the  failuie  of  groups  of  quantities 
taken  from  different  series  to  exhibit  an  orderly 
sequence.  Abundant  evidence  of  their  exist- 
ence has  been  encountered  in  various  discus- 
sions, including  the  control  by  conditions  of  the 
sensitiveness  of  capacity  to  slope,  of  the  sen- 
sitiveness of  capacity  to  form  ratio,  of  the 
value  of  the  optimum  form  ratio,  and  of  the 
value  of  the  constant  «;  but  the  present  dis- 
cussion is  the  only  one  affording  an  estimate  of 
their  magnitude. 

Assuming  that  the  estimate  of  2.5  per  cent 
represents  the  influence  of  a  restricted  class  of 
errors,  and  that  the  estimate  of  3.2  per  cent 
represents  the  joint  influence  of  that  class  and  a 
second  class,  the  independent  influence  of  the 
second  class  is  represented  by 


V3.22-2.52  =  ±  2.0  per  cent 

The  indication  is  that  the  two  classes  of  errors 
are  of  nearly  equal  importance.1 

These  results  are  qualified  by  the  fact  that 
the  estimate  of  3.2  per  cent  includes  not  only 
the  errors  of  the  adjusted  values  of  capacity  in 

1  The  considerations  making  it  probable  that  ±2.5  per  cent  is  an  over- 
estimate for  the  average  error  of  the  first  class  (p.  74)  do  not  apply  to  the 
estimate  of  ±  3.2  for  the  combined  error.  If  the  average  error  for  the  first 
class  is  as  low  as  ±2.0  por  cent,  the  computed  average  for  the  second 
class  becomes  ±2.5  percent. 


RELATION   OF   CAPACITY   TO   DISCHABGE. 


143 


Table  12,  but  also  whatever  errors  affect  the 
method  of  adjustment  in  connection  with  the 
discussion  of  the  control  of  capacity  by  dis- 
charge. That  method  includes  the  assump- 
tion that  capacity  varies  as  a  power  of  (Q  —  /c) 
and  also  certain  assumptions  as  to  the  control 
of  K  by  various  conditions.  If  the  errors  in- 
volved in  those  assumptions  could  be  elimi- 
nated or  discriminated,  the  general  estimate 
for  the  values  in  Table  12  would  be  somewhat 
reduced. 

As  the  chief  result  of  this  discussion,  the 
general  precision  of  the  main  body  of  material 
contributed  by  the  Berkeley  experiments  is 
characterized  by  an  average  probable  error 
slightly  in  excess  of  3  per  cent. 

This  estimate  applies  specifically  to  the  ad- 
justed capacities  of  Table  12  as  those  capacities 
are  related  to  slope  and  discharge,  and  it  can 
not  be  extended  to  derivatives  of  those  capaci- 
ties without  qualification.  It  is  believed  that 
the  precision  of  the  readjusted  capacities  of 
Table  32  is  higher,  and  also  that  of  the  values 
of  7,  in  Table  23,  but  that  the  values  of  ilt  jlt 
and  i3,  in  Tables  15,  16,  and  32,  rank  lower. 

In  the  adjustment  of  the  observations  on 
capacity  in  relation  to  slope,  and  also  in  the 
adjustment  of  observations  on  depth  and  slope, 
many  cases  were  treated  in  which  the  observa- 
tions were  either  not  sufficiently  numerous  or 
not  sufficiently  harmonious  to  afford  good  con- 
trol of  the  parameters  of  the  adjusting  equa- 
tions. In  such  cases  the  parameters  were  esti- 
mated in  groups,  with  orderly  sequences  of 
values.  A  similar  method  was  employed  also 
in  readjustments  in  relation  to  discharge. 
While  this  procedure  appeared,  and  still  ap- 
pears, to  be  the  best  practicable,  it  can  hardly 
fail  to  introduce  a  certain  amount  of  error. 
The  terms  of  the  adjusted  sequences  are  in- 
evitably associated  with  different  form  ratios; 
and  the  laws  connecting  form  ratio  with 
capacity  are  so  different  from  the  other  laws 
of  the  system  as  to  determine  sequences  less 
simple  than  those  actually  used.  The  ideal 
adjustment  would  take  simultaneous  cogni- 
zance of  the  complicated  interrelations  of 
capacity,  slope,  discharge,  and  form  ratio; 
but  such  comprehensive  treatment  can  not 
be  attempted  with  profit  until  we  have  a 
better  theoretic  knowledge  of  the  physical 
reactions. 


CONTROL    OP    RELATIVE    VARIATION    BY 
CONDITIONS. 

The  sensitiveness  of  capacity  to  changes  of 
discharge  is  indicated  by  i3,  the  index  of  rela- 
tive variation.  Inspection  of  the  values  of  that 
index  recorded  in  Table  32  shows  that  they 
vary  inversely  with  discharge  and  directly 
with  width,  and  both  these  tendencies  are  also 
to  be  inferred  from  the  lines  of  figure  47. 

The  table  further  indicates  that  the  rate  of 
change  in  the  index  in  response  to  change  of 
discharge  is  greater  for  small  discharges  than 
for  large.  This  feature  might  also  have  been 
inferred  from  the  plotted  lines,  but  its  system- 
atic expression  in  the  table  is  a  product  of  the 

formula,  i3  =  n      ,  by  which   the  values  were 
y  —  K 

computed. 

The  relation  of  the  index  to  width  of  channel 
is  not  similarly  dominated  by  the  formula, 
although  somewhat  influenced  by  the  assign- 
ment of  values  of  «.  The  variation  of  the 
index  with  width  is  in  general  more  pronounced 
for  low  discharges  than  for  high,  but  in  the 
data  for  high  discharges  occur  two  exceptions 
to  the  general  law.  These  exceptions  are 
ascribed  to  irregularities  in  the  data.  As  the 
width  for  any  particular  discharge  varies  in- 
versely with  the  form  ratio,  it  follows  that  the 
index  is  a  decreasing  function  of  form  ratio. 

The  relation  of  the  index  to  slope  is  not  shown 
by  the  table.  It  was  the  subject  of  a  special 
inquiry,  including  32  comparisons.  In  each 
comparison  a  value  of  the  index  for  a  particular 
slope  was  contrasted  with  the  value  for  a  slope 
twice  as  great,  the  other  conditions  being  the 
same.  In  25  instances  the  greater  index  was 
associated  with  the  smaller  slope;  in  7  instances 
with  the  larger.  The  mean  of  the  indexes 
computed  for  smaller  slopes  was  1.51;  the  mean 
for  larger  slopes  1.17.  The  general  law  appears 
to  be  that  the  index  varies  inversely  with 
slope.  The  seven  instances  of  opposite  tenor 
are  all  associated  with  large  discharges;  and 
their  occurrence  is  ascribed  to  a  systematic 
error  connected  with  the  assignment  of  values 
to  K. 

Table  35  compares  values  of  the  index  with 
fineness.  Despite  irregularities,  it  is  evident 
that  the  values  tend  to  increase  in  passing  from 
finer  to  coarser  grades — that  is,  their  variation 
in  respect  to  fineness  is  inverse. 


144 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


TABLE  35. —  Values  of  the  exponent  i3  arranged  to  show  vari- 
ation in  relation  to  fineness  of  debris. 


Grade. 

Values  of  i,. 

Q  =0.363 

Q-0.  182 

w«=0.  66 

w-1.00 

w-1.32 

w-0.66 

w=1.00 

(B 
(C 
(D 

(H 

1.17 
1.24 
1.06 
1.05 
2.04 
3.46 

1.21 
1.33 
1.24 
1.25 

1.33 
1.36 
1.44 
1.33 

1.28 
1.38 
1.21 
1.30 

1.43 

1.63 
1.64 
2.02 

In  summary,  the  sensitiveness  of  capacity  to 
variation  of  discharge  is  greater  as  slope,  dis- 
charge, form  ratio,  and  fineness  are  less. 


;,=/(£,<>, 


(68) 


DUTY  AND  EFFICIENCY. 


The  variation  of  capacity  with  discharge  is 
indicated  in  general  terms  by  an  equation  of 
the  type  of  (33) : 

ff-v.e*-.- (69) 

Duty  being  the  quotient  of  capacity  by  dis- 
charge, this  gives 

^=-. %L  =  v3Qi*-1--  -  (70) 


and,  as  efficiency  is  the  quotient  of  capacity  by 
discharge  and  slope, 


3i 

-  _?  O'*-1 

•  S  V 


(71) 


That  is,  the  index  of  relative  variation  for  both 
duty  and  efficiency,  in  relation  to  discharge,  is 
less  by  unity  than  the  corresponding  index  for 
capacity.  Therefore  the  values  of  the  index 
in  Table  32  need  only  to  be  reduced  by  unity 
to  apply  to  duty  and  efficiency. 

Under  ordinary  conditions  the  index  for  duty 
and  efficiency  falls  between  unity  and  zero ;  or, 
in  other  words,  duty  and  efficiency  increase 
with  increase  of  discharge,  but  their  increase  is 
less  rapid  than  that  of  discharge.  Exception- 
ally the  increase  is  much  more  rapid,  the  excep- 
tions being  associated  with  discharges  little 
above  the  limit  of  competence.  On  the  other 
hand,  there  appear  to  be  conditions  under  which 


the  index  falls  below  zero,  so  that  duty  and 
efficiency  diminish  with  increase  of  discharge. 
The  diminution  indicated  by  the  figures  in  the 
column  (of  Table  32)  for  grade  (D)  and  width 
0.66  foot  is  only  of  the  order  of  magnitude  of 
the  probable  error;  but  a  pronounced  diminu- 
tion would  be  inferred  from  the  values  of  the 
index  for  grade  (B)  and  width  0.23  foot.  As 
the  results  from  the  last-mentioned  group  of 
observations  stand  by  themselves  in  various 
respects,  some  reservation  is  felt  in  regard  to 
them,  and  there  is  at  least  room  for  doubt 
whether  the  diminution  is  actually  demon- 
strated. 

With  respect  to  all  conditions  the  variations 
of  the  index  for  duty  and  efficiency  follow  the 
same  laws  as  the  index  for  capacity;  but,  as  a 
consequence  of  the  uniform  reduction  by  unity, 
the  rates  of  variation  are  higher.  If,  for  ex- 
ample, in  passing  from  a  smaller  to  a  larger  dis- 
charge, the  index  for  capacity  falls  from  1.40  to 
1.20,  a  reduction  of  one-seventh,  the  index  for 
efficiency  falls  from  0.40  to  0.20,  a  reduction 
of  one-half. 

Lines  of  reasoning  strictly  parallel  to  those 
employed  in  the  last  section  of  Chapter  III 
yield  the  following  conclusions: 

The  synthetic  index  of  relative  variation  for 
the  duty  of  water  in  relation  to  discharge  is 
less  by  unity  than  the  corresponding  synthetic 
index  for  capacity  in  relation  to  discharge. 

The  synthetic  index  of  relative  variation  for 
efficiency  in  relation  to  discharge  is  less  by 
unity  than  the  corresponding  synthetic  index 
for  capacity  in  relation  to  discharge. 

If  the  duty  of  water,  or  if  efficiency,  be  as- 
sumed to  vary  as  some  power  of  Q  —  K,  the  ex- 
ponent of  that  power  (expressing  the  instanta- 

0  —  K 

neous  rate  of  variation)  equals  o  —    ~    • 

charge  increases  from  K  toward  infinity,  the 
exponent  diminishes  from  o  toward  o  —  1  . 

If  the  relation  of  efficiency  to  discharge  (  and 
similarly  for  the  relation  of  duty  to  discharge) 
be  expressed  by 

E^BAQ-Kj'n  ----------  (71a) 


As  dis- 


the  value  of  on  is  always  less  than  the  corre- 
sponding value  of  o,  the  difference  approaching 
but  not  exceeding  unity.  The  value  of  /q  is 


RELATION   OF   CAPACITY   TO   DISCHARGE. 


145 


always  greater  than  the  corresponding  value 
of  K,  usually  much  greater.  It  was  found  by 
trial  that,  within  the  range  of  conditions  real- 
ized in  the  laboratory,  the  difference  between 
values  of  efficiency  computed  directly  by  means 
of  (71a)  and  values  computed  indirectly  by 
means  of  ( 64)  is  not  large,  its  order  of  magnitude 
being  that  of  the  probable  errors. 

The  control  of  duty  and  efficiency  by  dis- 
charge is  further  considered  in  the  following 
section. 


COMPARISON  OF  THE  CONTROLS  OP  DIS- 
CHARGE AND  SLOPE. 

CONTROLS    OF    CAPACITY. 

We  are  now  in  position  to  compare  the  in- 
fluences exerted  by  slope  and  discharge,  sever- 
ally, on  capacity.  The  general  fact  brought  out 
by  the  comparison  is  that  capacity  is  more  sen- 
sitive to  changes  of  slope  than  to  changes  of 
discharge,  but  the  difference  in  sensitiveness  is 
not  the  same  for  all  conditions. 


TA  BLE  36. — -Comparison  of  the  index  of  relative  variation.i, .  for  capacity  and  slope,  with  the  index,  i3./or  capacity  and  discharge. 
[Foi1  grades  (A)  to  (D)  the  data  are  for  S— 1.0;  for  grades  (G)  and  (II)  for  S— 1.2.] 


Grade. 

Q 

10=0.44 

w—  0.66 

W=»1.00 

w=1.32 

w=1.96 

»i 

i> 

ii 

is 

ll 

k 

ii 

k 

ii 

is 

(A) 
(B) 

(C) 

(D) 

(0) 
(H) 

0.182 
.363 
.545 
.734 
1.119 

.093 
.182 
.363 
..545 
.734 
1.119 

.093 
.182 
.363 
.545 
.734 
1.119 

.093 
.182 
.363 
.545 
.734 

.363 
.734 
1.119 

.363 
.734 
1.119 

1.99 

1.28 

2.08 
1.83 

1.43 
1.21 

2.17 
1.70 

1.72 
1.33 

1.90 

1.41 

1.48 

1.14 

1.79 

1.12 

1.71 

1.20 

1.55 
1.31 

1.17 
1.10 

2.31 
1.81 

1.21 
1.10 

2.34 
1.82 
1.58 
1.49 

1.75 
1.38 
1.24 
1.20 

1.92 
1.67 
1.62 
1.61 

1.63 
1.33 
1.25 
1.21 

1.94 
1.75 
1.73 
1.71 

1.89 
1.36 
1.24 
1.19 

2.01 

1.87 
1.66 
1.60 

1.64 
1.40 
1.30 
1.21 

2.39 

1.88 

1.30 
1.14 

1.88 
1.73 
1.59 

1.58 
1.57 

1.68 
1.21 
1.06 
1.02 
1.00 

1.87 
1.59 
1.47 
1.45 
1.48 

1.64 
1.24 
1.15 
1.11 
1.07 

2.37 
1.67 
1.55 
1.47 

2.30 
1.44 
1.28 
1.21 

2.14 
1.87 
1.62 
1.85 

2.00 
1.60 
1.45 
1.32 

2.22 
1.80 

2.46 
2.33 

2.01 
1.69 
1.87 
1.65 

2.00 
1.22 
1.08 
1.02 

2.11 

1.33 

1.64 

1.01 

1.85 

1.01 

2.98 
2.42 
2.29 

6.03 
3.23 
2.78 

2.04 
1.29 
1.22 

3.46 

1.82 
1.57 

| 

1 

1 

In  Table  36  values  of  the  index  of  relative 
variation  are  brought  together  from  Tables  15 
and  32.  The  selection  includes  all  such  as  cor- 
respond in  respect  to  debris,  trough  width, 
slope,  and  discharge,  with  the  exception  of 
those  of  trough  width  0.23  foot,  which  appear 
to  be  anomalous.  There  are  64  pairs  of  values. 

Of  the  64  comparisons,  62  show  capacity  as 
more  sensitive  to  slope,  2  as  more  sensitive  to 
discharge.  The  two  exceptional  cases  are  from 
experiments  with  debris  of  grade  (D)  and  with 
channel  width  0.66  foot;  and  the  data  from 
those  experiments  were  reexamined  in  search 
for  an  explanation  of  what  seems  an  anomaly. 
No  explanation  was  found,  and,  as  the  observa- 
tions are  supported  by  the  estimates  of  pre- 
209210— No.  86 — 14 10 


cision,  it  remains  probable  that  there  are  real 
exceptions  to  the  general  rule. 

The  means  of  the  64  values  of  \  and  i3  are, 
severally,  1.93  and  1.42;  and  the  ratio  of  the 
first  to  the  second  is  1.36.  On  the  average, 
the  sensitiveness  of  capacity  to  slope  is  one- 
third  greater  than  the  sensitiveness  of  capacity 
to  discharge. 

• 

To   ascertain  the  variation  of  the  ratio  -»* 

H 

with  discharge,  the  values  in  Table  36  were 
specially  grouped  for  the  taking  of  partial 
averages.  The  first  group  gave  comparative 
ratios  for  discharges  of  0.093  and  0.182  ft.'/sec., 
by  means  of  four  sets  of  index  values,  each  set 
agreeing  as  to  all  conditions  other  than  dis- 


146 


TRANSPORTATION    OF   DEBRIS  BY   RUNNING   WATER. 


charge.  The  second  group  gave  comparative 
ratios  for  discharges  of  0.182,  0.363,  and  0.734 
ft.3/sec.,  by  means  of  five  sets  of  index  values; 
and  two  other  groups  made  other  comparisons, 
as  shown  in  Table  37.  The  upper  division  of 
the  table  gives  mean  values  of  it  and  ia;  the 


that  within  each  group  the  values  of  the 
indexes  decrease  as  discharge  increases,  while 
the  values  of  the  ratio,  as  a  rule,  increase  with 
the  increase  of  discharge.  To  the  first  rule 
there  are  no  exceptions;  the  exceptions  to  the 
second  are  not  so  important  as  to  leave  the 
principle  in  doubt. 


lower  division,  their  ratios.     It  will  be  observed 

TABLE  37. — •  Variations  of  the  indexes  it  and  13,  and  their  ratio,  in  relation  to  discharge. 


Croup. 

Sets. 

Q-0.093 

Q-0.182               Q-0.363 

Q-0.543 

Q-0.  734 

Q-  1.119 

ii 

b 

ii 

13 

ii 

I, 

1] 

is 

(l 

is 

ii 

is 

1 
2 
3 
4 

4 

5 
g 
7 

2.23 

1.48 

1.81 
2.02 

1.21 
1.74 

1.73 
1.76 
2.67 

1.30 
1.41 
1.97 

1.65 

1.16 

1.69 

1.26 

1.59 
1.97 

1.19 
1.32 

1.85 

1.22 

ii/t. 

«!/>! 

>:/•» 

<l/>8 

Wfc 

ii/i> 

1 
2 
3 
4 

1.50 

1.50 
1.16 

1.33 
1.25 
1.35 

1.42 
1.34 
1.49 

1.35 

1.52 

The  variations  of  the  indexes  with  discharge  as  a  factor  controlling  capacity,  is  more  pro- 
have  already  been  illustrated  in  another  way  nounced  for  large  discharges  than  for  small, 
(pp.  107,  143).  The  new  fact  brought  out  is  under  like  conditions  of  slope,  width,  and 
that  the  superiority  of  slope  over  discharge,  fineness. 

TABLE  38. —  Variations  of  the  indexes  it  and  i3,  and  their  ratio,  in  relation  to  width  of  channel. 


Group. 

Sets. 

10-0.44 

tc-0.66 

w-1.00 

10-1.32 

w-1.90 

I'l 

{3 

ii 

k 

I] 

k 

h 

h 

il 

13 

1 

2 
3 

4 
8 
8 

2.10 

1.19 

1.94 
1.67 

1.50 

1.17 

1.71 
1.63 

1.35 
1.20 

1.83 
1.66 

1.55 
1.28 

1.83 

1.50 

Ji/ii 

ii/is 

iilh 

ii/is 

ii/ij 

1 
2 
3 

1.77 

1.29 
1.40 

1.27 
1.36 

1.18 
1.30 

1.22 

A  different  grouping  of  the  index  values, 
but  similar  in  principle,  gave  the  means  and 
ratios  of  Table  38,  which  is  related  to  channel 
width  just  as  Table  37  is  related  to  discharge. 
With  a  single  exception,  the  mean  values  of 
indexes  increase  with  width;  thus  illustrating 
general  facts  previously  noted  on  pages  104 
and  143.  Without  exception,  the  ratios  of 
ij  to  is  decrease  with  increase  of  width.  The 
new  fact  brought  out  is  that  the  superiority  of 
slope  over  discharge,  as  a  factor  controlling 
capacity,  is  more  pronounced  for  narrow  chan- 
nels than  for  wide,  under  like  conditions  of 
slope,  discharge,  and  fineness. 

As  form  ratio  varies  inversely  with  width,  it 
follows  that  the  superiority  of  slope  is  more 


pronounced,  under  like  conditions,  when  form 
ratio  is  large  than  when  it  is  small. 

A  third  grouping  of  the  index  values,  making 
a  similar  comparison  of  their  averages  and 
ratios  with  fineness  of  debris,  is  reported  in 
Table  39  (p.  147).  The  mean  values  of  indexes 
increase  on  the  whole  in  passing  from  finer  to 
coarse  grades,  but  there  is  much  irregularity. 
The  same  irregularity  was  encountered  when 
these  relations  were  examined  in  other  con- 
nections. (See  pp.  108  and  143.)  The  ratios 
decrease  on  the  whole  from  finer  to  coarser, 
and  there  is  but  one  discordance  among  the 
sequences. 

To  compare  the  variations  of  the  indexes 
with  changes  in  slope,  the  32  pairs  of  values  of 


RELATION   OP   CAPACITY   TO   DISCHAKGE. 


147 


ia  mentioned  on  page  143  were  contrasted  with  of  the  three  groups  indicates  that  the  ratio 
corresponding  values  of  i,.  The  results  are  of  the  indexes  increases  with  increase  of 
summarized  in  Table  40,  below,  in  which  each  slope. 

TABLE  39. — -Variations  of  the  indexes  it  andi3,  and  their  ratio,  in  relation  to  fineness  of  debris. 


(J 

0 

(I 

») 

(< 

') 

(] 

>) 

« 

'•) 

(I 

r) 

1] 

fa 

ii 

i, 

ll 

ij 

ii 

h 

ll 

!3 

ll 

k 

1 

11 

1.77 

1.28 

1.74 

1.39 

1.76 

1.45 

2 

8 

1.79 

1.38 

1.65 

1.32 

1.87 

1.54 

3 

3 

2  70 

1  66 

4  63 

2  64 

I'l 

h 

!l 

ia 

til 

'3 

111 

k 

III 

k 

ill 

ii 

1 

1. 

38 

I. 

25 

1. 

21 

2 

1. 

30 

1. 

26 

1. 

21 

3 

1 

59 

1 

76 

TABLE  40. —  Variations  of  the  indexes  it  and  tj,  and  their  ratio,  in  relation  to  slope. 


Group. 

Sets. 

S-0.5 

S-1.0 

S-1.2 

S-2.0 

S-2.4 

ll 

fa 

ii 

fa 

ll 

is 

ii 

fa 

il 

ia 

1 
2 
3 

18 
10 
4 

1.90 

1.56 

1.6S 
1.87 

1.23 
1.45 

1.66 

1.12 

2.82 

1.47 

2.13 

1.08 

ll/fa 

ill's 

Jl/fa 

ii/ia 

ii/is 

1 
2 
3 

1.22 

1.35 
1.29 

1.49 

1.91 

1.98 

To  sum  the  results  of  the  preceding  para- 
graphs: The  sensitiveness  of  capacity  for  trac- 
tion to  changes  of  slope,  as  measured  by  the 
exponent  ilt  is  in  general  greater  than  its  sen- 
sitiveness to  changes  of  discharge,  measured 
by  ig.  The  superiority  of  the  control  by  slope 
persists  through  nearly  (or  perhaps  quite)  the 
entire  range  of  conditions  realized  in  the 
laboratory.  If  the  superiority  is  measured  by 

m 

the  ratio  ^,  its  average  value  (based  on  64  com- 

** 

parisons)  is  1  .36,  and  it  increases  with  increase 

of  slope,  discharge,  form  ratio,  and  fineness  of 
debris. 


(72) 


Another  mode  of  comparing  the  controls  by 
slope  and  discharge  is  by  means  of  the  syn- 
thetic index  of  relative  variation  (p.  99),  and 
in  some  respects  this  mode  is  more  satisfactory 
than  the  one  given  above.  The  synthetic  index, 
73,  of  the  relative  variation  of  capacity  and  dis- 
charge was  computed  for  21  cases,  in  each  of 
which  the  greater  discharge  was  approximately 


double  the  lesser.  The  synthetic  index  /„  of 
the  relative  variation  of  capacity  and  slope, 
was  computed  for  21  pairs  of  cases,  the  greater 
slope  being  double  the  lesser.  Each  value  of  7S 
was  joined  to  two  discharges  and  a  slope  and 
associated  with  a  pair  of  values  of  /,.  Each  of 
the  two  values  of  7t  was  joined  to  one  discharge 
and  two  slopes,  the  slopes  being  so  chosen, 
whenever  possible,  that  their  mean  coincided 
with  the  slope  of  the  associated  73.  The  mean 
of  the  42  values  of  7t  is  1.92;  that  of  21  values 
of  73  is  1 .42 ;  and  the  ratio  of  these  means  is  1 .35. 
The  result  is  practically  identical  with  that 
obtained  by  the  discussion  of  values  of  it  and  i3. 

CONTROLS    OF   DUTY. 

The  index  of  relative  variation  of  the  duty 
of  water  in  relation  to  slope  (p.  121)  is  ilr  the 
same  as  the  index  for  capacity  and  slope. 
The  corresponding  index  for  the  duty  of  water 
in  relation  to  discharge  (p.  144)  is  i3—  1.  The 

rt 

ratio  of  these  indexes,  - — —,  is  evidently  greater 
i3  — 

n 

than  the  ratio  -r,  which  has  just  been  consid- 


148 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


ered — that  is,  the  superiority  of  the  control  by 
slopes,  as  compared  with  the  control  by  dis- 
charge, is  more  strongly  marked  in  the  case  of 
duty  than  in  the  case  of  capacity. 

i.     1.93 
For   general    averages,    -  =  r-j-=  1.36,    and 

.  *t      _L93^1.3. 

^ 

In  Table  41  the  values  of  .  _^  ^  correspond  to 

those  of  t1  in  Tables  37  to  40— that  is,   they 

are  based  on  the  partial  means  of  those  tables 
and  are  arranged  under  the  same  groups,  the 
purpose  being  to  show  how  the  dominance  of 
control  by  slope,  as  expressed  by  a  ratio,  varies 
with  certain  conditions. 

TABLE  41. —  Variations  of  the  ratio  .    '     in  relation  to  dis- 
charge, width  of  channel,  fineness  of  debris,  and  slope. 


Values  of  ^—.  for  group  — 

1 

2 

3 

4 

Q-     0.  093 
.182 
.363 
.545 
.734 
1.119 

w—     0.  44 
.66 
1.00 
1.32 
1.96 

Grade  (A 
(B 
(C 
(D 
(0 
(I) 

S-         0.5 
1.0 
1.2 
2.0 
2.4 

4.6 
8.6 

2.7 

5.8 

4.3 

6.5 

2.8 

10.3 

8.4 

6.1 

8.4 

11.2 
3.9 

9.8 
4.9 
3.3 

8.1 
5.9 
3.7 

6.3 
4.5 
3.9 

8.0 
8.1 
3.5 

1 

4  9 

3.1 

3.4 

7.3 

4.2 

6.0 

13.8 



26.6 



In  comparing  this  table  with  the  tables  of 
-^,  the  most  conspicuous  feature  noted  is  that  the 


variation  of  — 


with  all  conditions  is  much 


more   pronounced    than    the   variation   of   A 

*3 

Associated  with  this  is  the  fact  that  the  excep- 
tions or  apparent  reversals  observed  in  Tables 
37  and  39  are  not  repeated  in  Table  41. 

In  verbal  generalization  of  the  tabulated 
results  it  is  to  be  borne  in  mind  (1)  that  the 
alphabetic  order  in  which  the  grades  are 
arranged  is  the  order  from  fine  to  coarse,  and 
(2)  that  variation  with  respect  to  form  ratio 


is  the  inverse  of  variation  with  respect  to 
width  of  channel.  The  general  fact  is  that 
the  dominance  of  control  by  slope,  as  com- 
pared with  control  by  discharge  —  a  dominance 
always  pronounced  —  is  notably  increased  by 
increase  of  slope,  discharge,  fineness,  or  form 
ratio. 


(73) 


CONTROLS    OF    EFFICIENCY. 

The  index  of  relative  variation  of  efficiency 
in  relation  to  slope  (p.  144)  is  il—  1;  and  the 
corresponding  index  for  efficiency  in  relation  to 
discharge  is  i3—l.  The  ratio  of  these  indexes 

n       "|  n 

^r  is  greater  than  •=*,  the  corresponding  ratio 

in  the  case  of  capacity,  with  exception  of  those 
doubtful  cases  in  which  il<i3.  That  is,  the 
superiority  of  the  control  by  slope,  as  com- 
pared to  the  control  by  discharge,  is  more 
strongly  marked  in  the  case  of  efficiency  than 
in  the  case  of  capacity.  It  is  also  true  that 

n      1  o 

-r<- — *-;-;  or  that  the  superiority  of  control 

l,n  1.  tro  ~~    -1 

by  slope  is  less  strongly  marked  for  efficiency 
than  for  duty.  For  the  general  averages 
•i,  =  1 .93  and  i,  =  1 .42  the  computed  values  of 
the  ratios  expressing  superiority  of  control  by 
slope  are,  as  to  capacity  1.36,  as  to  efficiency 
2.21,  as  to  duty  4.3. 

TABLE  42. — -Variations  of  the  ratio  ^T  in  relation  to  dis- 
charge, width  of  channel,  fineness  of  debris,  and  slope. 


Values  of  .'—  j  for  group  — 

1 

2 

3 

4 

Q  —     0  093 

2.6 
3.9 

.182 
.363 
.545 
.734 
1.119 

tt>=     0.  44 
.66 
1.00 
1.32 
1.96 

Grade  (A) 

(B) 
(C) 
(D) 
(0) 
(H) 

S=         .05 
1.0 
1.2 
2.0 
2.4 

1.4 
2.4 

1.8 
2.7 
3.1 

1.7 

4.1 

3.0 
3.9 

5.5 
1.9 

3.9 
2.0 
1.5 

3.1 
2.4 

1.7 

2.7 
1.9 

1.7 

3.6 
2.0 
1.6 

3  0 

2.4 

1.6 
2.9 

1.9 

3.9 

.      ..1       fi-fi 



14:1 



EELATION    OF   CAPACITY   TO   DISCHARGE. 


149 


i  —  1 
In  Table  42  the  values  of  -^  —  =-  correspond  to 

*3~ 

n 

those  of  .  •  1  ,-  in  Table  41   and  are  similarly 

\~ 
derived  from  data  of  Tables  27  to  40.     The  pur- 

pose of  the  table  is  to  show  how  the  dominance 
of  control  by  slope,  aa  expressed  by  a  ratio, 
varies  with  certain  conditions. 

A  comparison  of  tabulated  values  for  the 
several  ratios  shows  that  the  ratios  associated 
with  efficiency  vary  with  conditions  more  rap- 
idly than  those  associated  with  capacity,  but 
less  rapidly  than  those  associated  with  duty. 
In  Table  42,  just  as  in  Table  41,  there  are  no 
exceptions  as  to  the  direction  of  the  trend  of 
variation. 

Bearing  in  mind  that  the  alphabetic  order  in 
which  the  grades  are  arranged  is  the  order  from 
fine  to  coarse,  and  that  variation  with  respect 
to  form  ratio  is  the  inverse  of  variation  with 
respect  to  width,  we  see  that  the  general  fact 
shown  by  the  table  is  that  the  dominance  of 
control  by  slope  —  a  dominance  always  pro- 
nounced —  is  notably  increased  by  increase  of 
slope,  discharge,  fineness,  or  form  ratio. 


- 


(74) 


SUMMARY. 

With  debris  of  a  particular  size  and  a  chan- 
nel bed  of  a  particular  slope,  there  is  a  particu- 
lar discharge  which  is  barely  competent  to 
cause  transportation.  With  increase  of  dis- 
charge above  this  barely  competent  disc  harge, 
there  is  a  proportional  addition  to  the  stream's 
potential  energy.  The  relation  of  capacity  to 
discharge  is  formulated  on  the  assumption  that 
the  capacity  is  proportional  to  some  power  of 
the  added  energy,  and  therefore  to  the  same 
power  of  the  added  discharge.  As  each  grade 
of  debris  is  somewhat  heterogeneous  as  to  the 
size  of  its  grains,  this  assumed  principle  can  not 
be  applied  strictly;  the  practical  assumption  is 
that  capacity  varies  with  a  power  of  the  differ- 
ence between  the  discharge  and  a  constant 


discharge,  the  constant  being  so  chosen  as  best 
to  harmonize  the  data. 

By  means  of  such  formulation  the  data  were 
readjusted  and  the  rate  of  variation  of  capacity 
with  discharge,  or  the  index  of  relative  varia- 
tion, i,,  has  been  computed  for  a  variety  of 
conditions.  It  is  found  to  be  greater  as  the 
slope  of  channel,  the  discharge,  the  fineness  of 
debris,  and  the  form  ratio  are  less.  The  aver- 
age of  the  values  computed  lor  laboratory  con- 
ditions is  1 .42  and  the  ordinary  range  is  from 
1.00  to  2.00. 

The  rate  at  which  the  efficiency  of  the  stream 
and  the  duty  of  the  stream's  water  vary  with 
discharge  is  denoted  by  an  index  which  is  less 
by  unity  than  that  for  capacity.  Its  average 
is  0.42  and  its  ordinary  range  is  from  0  to  1.00. 

It  has  previously  been  shown  that  the  corre- 
sponding indexes  showing  the  relation  of 
capacity  to  slope  are  larger.  In  other  words,  ' 
capacity  is  more  sensitive  to  changes  of  slope 
than  to  changes  of  discharge.  If  relative  sensi- 
tiveness to  the  two  controls  be  expressed  by  a 
ratio,  the  average  value  of  that  ratio  is  1.36. 
The  ratio  varies  with  conditions,  being  rela- 
tively large  when  slope,  discharge,  fineness, 
and  form  ratio  are  relatively  small. 

The  primary  adjustment  of  observations  of 
capacity,  described  in  Chapter  II,  was  an  ad- 
justment with  respect  to  slope.  The  probable 
errors  computed  from  differences  between  ad- 
justed and  unadjusted  values  were  influenced 
by  only  a  portion  of  the  observational  errors. 
In  readjusting  values  of  capacity  with  respect 
to  discharge,  another  division  of  the  observa- 
tional errors  was  encountered  and  its  import- 
ance was  estimated.  The  probable  errors 
computed  from  the  results  of  the  two  adjust- 
ments are  believed  to  represent  with  sufficient 
approximation  the  order  of  precision  of  the  ad- 
justed values  of  capacity,  which  constitute  the 
main  body  of  data  for  the  discussions  of  the 
report.  The  order  of  precision  is  expressed  by 
saying  that  the  average  probable  error  of  the 
adjusted  values  is  a  little  more  than  3  per  cent. 


CHAPTER  VI.— RELATION  OF   CAPACITY  TO   FINENESS  OF  DEBRIS. 


FORMULATION. 

To  study  the  laws  affecting  the  control  of  ca- 
pacity for  traction  by  fineness  of  debris,  capaci- 
ties should  be  compared  which  are  subject  to 
like  conditions  in  all  other  respects.  For  this 
purpose  data  from  Table  12  were  arranged  as 
in  Table  43,  where  the  capacities  in  each  hori- 
zontal line  are  conditioned  by  the  same  slope, 
discharge,  and  width  of  channel.  Ah1  the  data 
of  that  table  pertain  to  a  slope  of  1.0  per  cent; 
but  similar  tables  were  constructed  for  slopes 
of  0.5,  0.7,  1.4,  and  2.0  per  cent. 

The  same  data  were  also  plotted  on  logarith- 
mic paper;  and,  after  a  preliminary  examina- 
tion, five  sets  were  selected  for  special  investi- 
gation. The  plots  of  these  appear  in  figure  50, 
where  ordinates  are  logarithms  of  capacity  and 
abscissas  are  logarithms  of  linear  fineness.  It 
is  to  be  noted  that  the  zero  of  log  C  is  not  the 
same  for  the  different  graphs.  The  graphs  were 
moved  up  or  down,  so  as  to  avoid  confusion 
through  intersection. 

The  first  law  illustrated  by  the  plots  is  that 
capacity  increases  as  fineness  increases;  the 
second,  that  it  increases  more  rapidly  for  small 
fineness  than  for  great  fineness.  Despite  ir- 
regularities of  the  data  it  is  evident  that  the 


locus  of  log  C=f  (log  F)  is  a  curve,  and  that  the 
function  has  a  general  resemblance  to  those 


fjfj    (G>      (F) 


(B)fA) 


FIGUKE  50.— Logarithmic  plots  of  capacity  for  traction  in  relation  to 
fineness  of  de'bris;  corresponding  to  data  in  Table  44. 

found  in  comparing  capacity  with  slope  and 
discharge. 


TABLE  43. —  Values  of  capacity  for  traction,  arranged  to  illustrate  the  relation  of  capacity  to  grades  of  debris. 


Conditions. 

Value  of  C  for  grade— 

S 

w 

Q 

(A) 

(B) 

(C) 

(D) 

(K) 

CD 

(G) 

(H) 

1.0 

0.66 
1.00 
1.32 
1.96 

0.093 
.182 
.363 
.545 
.734 
1.119 

.093 
.182 
.363 
.545 
.734 
1.119 

.093 
.182 
.363 
.545 
.734 
1.119 

.093 
.182 
.363 
.545 
.734 
1.119 

10.8 
33.5 
81 
120 

13.8 
32.1 
73 
112 
152 

9.1 

29.8 

39.5 

24.8 

20.5 

8.5 

140 

101 

40 

39 

30.4 
49 

12.8 
25.2 

37.5 
100 

30.1 
85 
143 
199 

31.2 
85 
140 
180 
276 

24.0 
73 
108 
153 

14.8 
33.8 

14.0 

231 

72 
138 

42.8 
79.5 

26.9 
61 

36.8 
104 

29.3 
79 
140 
204 

21.4 
79 
129 
187 

59.2 

36.3 

250 

131 

73 
123 

40.2 

70 

18.9 
52.2 

96 

67.  « 
120 
190 
313 

60.0 
111 
190 
343 

240 
359 

150 


KELATION   OF   CAPACITY   TO   FINENESS   OF   DEBRIS. 


151 


For  any  velocity,  as  determined  by  slope, 
discharge,  and  width,  there  is  a  competent 
fineness,  marking  the  limit  between  traction 
and  no  traction ;  and  to  this  extent,  at  least,  the 
relation  of  fineness  to  traction  is  analogous  to 
the  relations  of  slope  and  discharge.  It  is  not 
easy  to  carry  the  analogy  further,  because  slope 
and  discharge  are  conditions  of  active  force, 
and  fineness  is  a  condition,  of  reactive  force,  or 
resistance;  but  an  experiment  in  formulation 
reveals  a  parallelism  quite  as  striking  as  that 
between  the  capacity-slope  and  capacity-dis- 
charge functions. 


Assuming 


-(75) 


in  which  0  is  a  constant  fineness  and  bt  a  con- 
stant of  the  numerical  value  of  capacity  when 
F=  <£  +  !,'  the  five  sets  of  data  in  Table  44 
were  treated  graphically  for  the  determination 
of  <j>  and  p.  The  methods  were  such  as  have 
already  been  described  (pp.  65  and  139).  The 
formulas  were  then  used  to  compute  read- 
justed values  of  capacity,  Cr,  and  values  of  the 
index  of  relative  variation,  it>  and  probable 
errors  were  also  computed. 


TABLE  44. — Numerical  data  connected  with  the  plots  in  figure  SO,  and  illustrating  the  relation  of  capacity  for  traction  to  grades 

of  debris. 


1 

2 

3 

\w  
Conditions  of  experiments  <Q  

Is  

0.66 
0.363 
1.4 

1.00 
0.734 
0.7 

1.00 
0.363 
1.4 

(4> 

41 
0.62 
4.5 

55 
0.60 
2.4 

49 
0.58 
3.7 

p  

f  C 

6.8 
2.8 

7.6 
2.9 

4.9 
1.9 

C           C,            tt 

ft 

C 

ft 

'. 

ii 

C 

Cr 

i« 

ii 

Grade  (\) 

121 
110 
106 
85 
40 
20 
8 

140 
123 
102 
76 
42 
21.7 
7.4 

0.63 
.64 
.66 
.70 
.87 
1.42 
5.45 

1.83 
1.66 
1.50 
1.72 
1.68 
2.35 
4.11 

185 
149 
143 
127 
62 
33.4 
16.3 

196 
173 
143 
108 
62 
34.4 
16.3 

0.61 
.62 
.63 
.66 
.80 
1.18 
2.80 

1.79 
1.63 
1.53 
1.63 
1.78 
2.42 
3.15 

Grade(B). 
Grade  (C). 

137           145             0.65 
124          123              .  67 
.69 

1.54 
1.55 

Data  and  computed  results  ....  Q^e  (  E) 
Grade  (F)! 
Grade  (G). 
Grade  (H). 

49.4        59              .81 
41            37            1.08 
23            22            1.82 
6.  9          7.  1       13.  50 

2.10 
1.99 
2.71 
4.69 

1 

4 

\w  
Conditions  of  ex  periments  <  Q  

Is  

1.32 
0.734 
1.0 

1.32 

0.363 
2.0 

Parameters  of  equations  

f*  

fe::::: 

48 
0.61 
3.8 

55 
0.50 
11.4 

1C  
(Or  

1.7 
0.6 

6.2 
2.4 

C 

C, 

i. 

ii 

C 

C, 

i, 

k 

Gra 

Gra 
Gra 

le(A). 
le(B). 
3e  (C)  . 
3e(D). 
ie(E). 
le(F). 
ie(G). 
ie(IJ).. 

250 
204 
187 
131 
73 
40.2 
18.9 

250 
219 
180 
133 
74 
40.2 
19 

0.64 
.65 
.66 
.70 
.84 
1.22 
5.20 

1.71 
1.71 
1.47 
1.85 
1.54 
2.40 
3.43 

328 
255 
233 
237 
112 
66 
28.3 

319 
285 
242 
190 
115 
67 
27.4 

0.53 
.54 
.55 
.58 
.72 
1.19 
4.55 

.97 
.63 
.52 
.92 
.60 
2.03 
3.67 

Gra 
Gra 
Gra 

PRECISION. 


The  average  probable  error  of  the  read- 
justed capacities  was  found  to  be  ±2.1  per 
cent.  This  error  is  to  be  ascribed  in  part  to 
discordance  of  the  data  among  themselves,  and 


in  part  to  discordance  of  the  formula  with  the 
data;  but  the  distribution  of  the  residuals  is 
not  such  as  to  imply  important  discordance  of 
the  formula. 

i  The  dimensions  of  capacity  are  -M+i  T-',  and  of  fineness  £-*.    The 
constant  6(,  being  equal  to  C/  (F—*t>)p,  hag  dimensions  L+'P  J/+'  T~l. 


152 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATE&. 


There  are,  however,  important  discordances 
among  the  data.  Considered  as  errors,  the 
discordances  constitute  a  group  which  were  not 
detected  in  the  adjustments  of  capacities  to 
slopes  and  to  discharges,  but  which  escaped 
those  tests  because  related  peculiarly  to  the 
grades  of  debris.  From  the  residuals  of  the 
present  readjustment  the  average  probable 
error  of  capacities  before  readjustment  is  esti- 
mated at  ±5.4  per  cent,  whereas  the  average 
probable  error  of  the  body  of  once-adjusted 
capacities  from  which  these  were  selected  was 
estimated  at  2.5  per  cent.  On  the  assumption 
that  the  estimate  for  the  whole  body  of  values 
applies  to  the  selected  group,  the  share  of  error 
associated  with  the  grades  is  estimated  as 

V5.42-2.52=  ±4.8  per  cent. 

Inspection  of  the  logarithmic  plots  suggested 
that  part  of  the  discordance  of  the  data  is  syste- 
matic. To  bring  out  the  systematic  element 
the  original  values  of  capacity  in  Table  44  were 


divided  by  the  readjusted  values,  and  means 
were  taken  of  the  quotients.  The  means  are 
listed  below,  and  in  figure  51  they  are  plotted 
logarithmically  in  relation  to  fineness.  The 
plotted  points  conspicuously  out  of  line  are 
those  for  grades  (B)  and  (D),  the  capacities 
determined  for  grade  (B)  being  relatively  too 
small  and  those  for  grade  (D)  too  large.  The 
same  result  was  obtained  from  a  canvass  of  a 
wide  range  of  data. 

Ratios  oj  original  values  of  capacity   to   adjusted  values. 


Grade  .  .. 

(A) 

(B> 

(C) 

(D) 

(E) 

(F) 

(G) 

1  002 

812 

602 

388 

178 

95.9 

43.4 

0.96 

0.89 

1.01 

1.13 

0.98 

0.97 

1.02 

It  is  surmised  that  these  errors  arise  in  part 
from  variations  of  experimental  method,  and 
this  suspicion  attaches  especially  to  grade  (B), 
which  was  the  first  to  be  treated  in  the  labora- 
tory. It  attaches  much  less  to  grade  (D),  for 


(G) 


(E) 


CD) 


(C)       CB)    (A) 


FIGURE  51.— Average  departures  of  original  values  of  capacity  from  the  system  of  values  readjusted  in  relation  to  fineness  of  d<Sbris.    Thehort 
lontal  line  represents  the  readjusted  system.    The  broken  lines,  above  and  below,  correspond  to  departures  of  10  per  cent. 


which  the  experimental  method  was  about  the 
same  as  for  grades  (A),  (E),  and  (F).  So  far  as 
the  quality  of  the  experimental  work  is  gaged 
by  the  probable  error,  that  on  grade  (D)  would 
appear  to  be  considerably  below  the  average. 

It  is  surmised  that  systematic  error  may  also 
be  connected  with  properties  of  the  grades 
other  than  that  of  fineness.  Range  of  fineness 
has  already  been  appealed  to  as  an  explanation 
of  apparent  anomalies  as  to  competent  slope. 
From  the  experiments  with  mixtures  (Chapter 
IX)  we  know  that  great  range  within  a  grade 
would  tend  to  increase  capacity  for  traction. 
As  the  range  is  small  for  grade  (A),  regularly 
increases  to  grade  (E),  and  is  again  small  from 
(F)  to  (H),  the  influence  of  this  factor  tends  to 
enhance  the  capacity  of  grade  (E) ;  but  the 
effect  of  that  influence  is  not  apparent  in  the 
diagram. 

The  separation  of  grades,  being  effected  by 
sieves,  was  a  gaging  of  grains  by  certain  dimen- 
sions. The  grains  may  have  differed  also  as  to 
shapes  and  densities,  and  each  of  these  proper- 


ties would  affect  capacity.  The  debris  first 
obtained  for  the  laboratory  had  been  washed 
from  the  bed  of  Sacramento  River  when  a  flood 
broke  the  levee  below  the  mouth  of  American 
River.  This  furnished  material  for  the  finer 
grades.  Other  lots  of  debris  were  taken  from 
the  bed  of  American  River,  and  these  furnished 
material  chiefly  for  the  coarser  grades.  So  far 
as  the  separations  from  the  different  lots  coin- 
cided, they  were  used  indiscriminately.  It  is 
possible  that  grade  (D)  was  composed  chiefly 
of  the  coarser  particles  of  a  fine  alluvium,  while 
grade  (E)  was  composed  chiefly  of  the  finer 
particles  of  a  relatively  coarse  alluvium.  As 
any  natural  body  of  tractional  debris  is  the 
result  of  a  sorting  process  in  which  tractionable 
particles  are  separated  from  the  immovable  on 
one  side  and  from  the  suspendible  on  the  other, 
it  is  likely  to  include  among  its  coarser  grains 
many  which  are  tractionable  only  because  of 
low  density  or  favorable  shape,  and  among  its 
finer  grams  many  which  escape  suspension 
because  of  high  density  or  unfavorable  shape. 


RELATION   OF   CAPACITY   TO   FINENESS   OF   DEBBIS. 


153 


Thus  it  appears  possible  that  the  superior 
mobility  of  grade  (D)  was  determined  by 
properties  other  than  size.  Unfortunately  the 
record  is  not  of  such  character  that  the  value 
of  this  suggestion  can  now  be  tested. 

A  third  suggestion  pertains  to  the  gaging  of 
fineness.  The  method  of  gaging  included  a 
weighing  and  involved  certain  assumptions  as 
to  homogeneity  in  average  density  and  in  shape 
which  may  not  have  been  fully  warranted. 

These  various  suggestions,  while  not  suscep- 
tible of  test  at  the  present  time,  are  sufficiently 
plausible  to  show  the  possibility  of  definite 
causes  for  the  discordances  discovered  by  the 
comparison  of  data  from  different  grades.  In 
my  judgment  it  is  proper  to  ascribe  the  greater 
discordances  to  such  causes,  and  to  view  them 
as  abnormalities  with  respect  to  the  law  con- 
necting capacity  with  fineness. 

In  view  of  the  magnitude  of  the  abnormalties 
or  discordances,  it  does  not  appear  profitable 
to  extend  the  readjustment  of  data  to  other 
and  shorter  sets.  The  five  sets  in  Table  44 
were  selected  because  they  included  great  range 
in  fineness,  and  because  they  were  qualified  to 
yield  fairly  definite  values  of  the  constant  </>. 

VARIATIONS  OF  THE  CONSTANT  <f>. 

The  laws  which  control  the  variation  of  0 
have  not  been  developed  from  the  observations, 
but  their  general  character  may  be  inferred 
deductively  by  considering  the  relations  of  com- 
petent fineness  to  various  conditions  —  it  being 
assumed  that  <J>  is  intimately  related  to  com- 
petent fineness.  Postulate  a  current  of  which 
the  velocity  is  determined  by  a  particular  slope, 
a  particular  discharge,  and  a  particular  width. 
For  this  current  a  certain  fineness  is  competent. 
Increase  of  slope  or  discharge  increases  the 
velocity  and  makes  a  lower  fineness  competent. 
Decrease  of  width,  which  corresponds  to  in- 
crease of  form  ratio,  increases  velocity  and 
makes  a  lower  fineness  competent.  Thus  com- 
petent fineness,  and  therefore  <f>,  varies  inversely 
with  the  slope,  discharge,  and  form  ratio. 

A)  ________  --(76) 


capacity  and  fineness.     The  formula  for  the 
index  (cf.  pp.  100  and  141)  is 


-(78) 


INDEX  OF  RELATIVE  VARIATION. 

Framing  an  equation  of  the  type  of  (33)  — 

C=viF{<  .............  (77) 

in  which  it  is  the  index  of  relative  variation  for 


With  this  formula,  values  of  it  were  computed 
from  the  data  in  Table  44,  and  they  are  given 
in  the  lower  part  of  that  table. 

By  inspection  it  appears  that  the  index  in- 
creases as  fineness  diminishes,  its  growth  being 
at  first  slow  but  becoming  rapid  as  competent 
fineness  is  approached.  Because  of  the  dis- 
cordances of  the  data  it  is  not  easy  to  derive  a 
body  of  values  of  the  index  for  discussion  in 
relation  to  other  conditions,  but  it  is  relatively 
easy  to  obtain  comparative  values  of  the  syn- 
thetic index,  It,  and  the  variations  of  these 
values  may  be  assumed  to  show  the  same  trends 
as  the  variations  of  it.  Values  of  74  were  com- 
puted between  corresponding  data  of  grades 
(C)  and  (G)  by  the  formula 

j     log  Cl- 


-logtf- 

in  which  (7,  and  Ca  are  specific  capacities  corre- 
sponding to  the  finenesses  F,  and  Fn;  and  the 
results  are  given  in  Table  45. 

From  these  results  it  is  inferred  (1)  that  the 
index  varies  inversely  with  the  slope,  (2)  that 
it  varies  inversely  with  discharge,  and  (3)  that 
it  varies  directly  with  width,  and  therefore 
inversely  with  form  ratio.  The  response  is  in 
general  of  a  very  pronounced  character,  but  to 
this  there  is  exception  in  one  of  the  compari- 
sons with  width.  It  is  possible  that  the  index 
is  a  maximum  function  of  width  and  a  mini- 
mum function  of  form  ratio.  With  some  reser- 
vation on  this  point,  we  may  generalize: 

1,  P,  &)-.  -.(79) 


If  equation  (79)  be  compared  with  equations 
(39)  and  (68),  it  will  be  seen  that  the  variation 
of  the  capacity-fineness  index  observes  the  same 
laws  of  trend  as  the  variations  of  the  capacity- 
slope  index  and  the  capacity-discharge  index. 
In  view  of  this  general  parallelism  of  variation, 
it  is  thought  that  the  relative  magnitudes  of 
average  it,  average  i,,  and  average  i3  may  be 
adequately  discussed  by  means  of  a  moderate 
number  of  comparisons.  Accordingly  only 
those  values  of  it  computed  from  the  five  equa- 
tions of  Table  44  are  used.  The  corresponding 
values  of  i3  are  not  available,  but  those  of  i, 


154 


TRANSPORTATION    OF    DEBRIS   BY    RUNNING    WATER. 


are  given  in  Table  15.  They  have  been  copied 
into  Table  45,  so  as  to  be  conveniently  com- 
pared. 

Inspection  shows  that  it  is  in  general  much 
smaller  than  \  but  that  it  becomes  greater  as 
the  limit  of  competence  is  approached.  As  to 
the  first  of  these  generalizations  there  can  no 
question,  but  the  second  is  not  equally  satis- 
factory. In  the  vicinity  of  competence  the 
value  of  it  is  highly  sensitive  to  the  influence 
of  <f>;  and  in  the  same  region  \  is  highly  sensi- 
tive to  a.  The  features  of  the  table  might  be 
produced  by  slight  overestimates  of  <f>  or  by 
slight  underestimates  of  a.  In  view  of  this 
consideration  it  is  probably  best  to  leave  the 
higher  values  of  the  index  out  of  the  account 
and  base  a  computation  of  averages  wholly  on 
the  lower  values.  Including  only  the  28  values 
of  each  index  associated  with  grades  (A)  to  (F), 
the  means  are,  for  it  0.77,  for  it  1.79;  and  the 
ratio  of  the  second  to  the  first  is  2.4;  that  is, 
the  sensitiveness  of  capacity  to  control  by  slope 
is  estimated  to  be  2.4  times  as  great  as  its  sen- 
sitiveness to  control  by  fineness.  The  ratio  of 


sensitiveness  for  slope  and  discharge  iji3  having 
been  estimated  at  1.36,  it  follows  that  the  ratio 

2  4 
for  discharge  and  fineness  is  =-55  —  1.8. 

Mean  it  :  mean  is  :  mean  it  :  :  2.4  :  1.8  :  1.0. 

It  is  to  be  understood  that  these  estimates 
are  of  the  most  general  character.  The  ratios 
doubtless  vary  notably  with  conditions. 

The  property  with  which  capacity  has  been 
compared  in  this  chapter  is  linear  fineness,  F, 
defined  as  the  reciprocal  of  diameter,  or  as  the 
number  of  grains  to  the  linear  foot.  Bulk  fine- 
ness, F2,  defined  as  the  reciprocal  of  volume,  is 
proportional  to  the  cube  of  linear  fineness.  It 
follows  that  the  index  of  relative  variation 
when  capacity  is  compared  with  bulk  fineness 
is  one-third  the  corresponding  index,  it,  for 
capacity  and  linear  fineness;  and  the  same 
factor  applies  to  synthetic  indexes.  If  bulk 
fineness  were  substituted  for  linear  fineness  in 
equations  of  the  form  of  (75),  the  values  of  <f> 
would  be  quite  different  and  the  values  of  p 
would  be  uniformly  one-third  as  great. 


TABLE  45.  —  -Values  of  It,  the  synthetic  index  of  relative  variation  for  capacity  and  fineness,  compared  with  slope,  discharge, 
and  width  of  channel. 

Fixed  conditions  

Coarser  grade  

(G) 
(C) 

(G) 
(C) 

if 

(G) 
& 

(G) 
(C) 
2.0 
0.363 

(G) 
(C) 
2.0 
0.734 

Discharge  (ft.s/sec.)  .  . 
Width  (feet)  

0.363 
1.32 

0.734 
1.32 

1.32 

1.32 

8 

/, 

S 

It 

Q 

It 

Q 

A 

w 

/( 

w 

/, 

1.8 
2.4 

1.05 

.76 

1.0 
2.0 

0.99 
.61 

0.363 
.734 

1.05 
.65 

0.363 
.734 

0.92 

.61 

0.66 
1.00 
1.32 

0.59 

.74 
.92 

1.00 
1.32 

0.57 
.61 

DUTY  AND  EFFICIENCY. 

The  relations  of  duty  and  efficiency  to  ca- 
pacity involve  discharge  and  slope  but  are  in- 
dependent of  fineness.  Fineness,  therefore,  has 
exactly  the  same  control  of  duty  and  efficiency 
that  it  has  of  capacity,  and  the  conclusions  of 
this  chapter  apply  without  qualification  to  duty 
and  efficiency. 

SUMMARY. 

Capacity  for  traction  is  greater  for  fine  d6bris 
than  for  coarse — that  is,  capacity  increases  with 
fineness.  The  law  of  increase  admits  of  formu- 
lation in  a  manner  strictly  analogous  to  that 
employed  in  comparing  capacity  with  slope 
and  discharge — that  is,  it  is  found  that  ca- 
pacity varies  approximately  with  a  power  of 
the  fineness  less  a  constant  fineness.  The  value 


of  the  constant  finenes3  varies  with  conditions, 
being  greater  as  slope,  discharge,  and  form 
ratio  are  greater.  The  rate  at  which  capacity 
varies  with  change  of  fineness,  or  the  index  of 
relative  variation,  is  not  the  same  for  all 'con- 
ditions, being  greater  as  slope,  discharge  and 
form  ratio  are  less.  Under  similar  conditions 
the  rate  is  less  than  the  corresponding  rate  for 
capacity  and  slope,  the  average  ratio  between 
them  being  as  1  to  2.4. 

The  arrangement  of  capacities  in  accordance 
with  the  assumed  law  of  increase  develops  dis- 
crepancies which  are  believed  to  be  of  the  na- 
ture of  systematic  errors.  The  largest  of  these 
have  a  magnitude  of  about  10  per  cent.  They 
are  tentatively  ascribed  to  peculiarities  of  the 
<16bris  used  in  experiments  and  to  imperfectly 
developed  laboratory  methods. 


CHAPTER  VII.— RELATION  OF  CAPACITY  TO  VELOCITY. 


PRELIMINARY  CONSIDERATIONS. 

The  work  of  stream  traction  is  accomplished 
by  the  movement  of  water  along  the  bed  of  the 
channel.  For  that  reason  the  system  of  water 
movements  and  water  velocities  near  the  bed 
is  intimately  related  to  the  load  or  capacity  for 
load.  In  certain  parts  of  this  paper  and  in 
the  writings  of  some  other  investigators  use  is 
made  of  the  term  "bed  velocity,"  or  its  equiva- 
lent, but  the  term  has  no  satisfactory  defini- 
tion. The  difficulties  which  are  encountered 
in  this  connection  have  to  do  also  with  the 
vertical  velocity  curve. 

In  all  the  streams  with  which  we  are  here 
concerned  the  flow  is  eddying  or  turbulent. 
At  any  point  the  direction  of  motion  and  the 


FIGITRE  52.— Vertical  velocity  curve,  drawn  to  illustrate  its  theoretic 
character  near  the  stream's  bed.    OD  is  the  origin  of  velocities. 

velocity  are  constantly  changing.  If  a  mean 
be  taken  of  the  instantaneous  forward  com- 
ponents of  velocity — the  components  parallel 
to  the  axis  of  the  stream — it  gives  for  the 
point  a  mean  velocity  coordinate  with  the 
mean  velocity  for  the  cross  section  obtained 
by  dividing  the  discharge  by  the  sectional  area. 
It  will  bo  observed  that  the  mean  at  a  point  is 
a  mean  with  respect  to  time,  while  the  sec- 
tional mean  is  primarily  a  mean  with  respect  to 
space.  The  mean  at  a  point,  as  thus  defined, 
being  called  Vp,  the  vertical  velocity  curve 
may  be  defined  as  the  curve  obtained  by  plot- 
ting the  values  of  Vp  for  any  vertical  of  the 
current  in  their  relation  to  depth.  As  commonly 
drawn  by  hydraulic  engineers,  it  terminates 
downward  at  some  distance  from  the  origin  of 
velocities,  OD — say  at  B  in  figure  52 — connoting 
a  finite  velocity  for  the  water  in  actual  con- 


tact with  the  bed.  This  implication  contra- 
venes a  theorem  of  hydrodynamics  that  the 
velocity  at  contact  with  the  wall  of  a  con- 
duit is  either  zero  or  indefinitely  small.  The 
theorem  is  believed  to  have  been  established 
experimentally  by  the  work  of  J.  L.  M. 
Poiseuille  *  and  is  generally  accepted.  In  the 
direct  study  of  the  velocities  of  streams  instru- 
mental observation  is  not  carried  from  surface 
to  bed,  but  ceases  at  some  point,  C,  and  the 
drawing  of  the  curve  below  that  point  is  a 
matter  of  inference.  The  inference  accordant 
with  the  hydrodynamic  principle  is  that  the 
curve  changes  its  course  below  C  and  reaches 
the  origin  at  or  near  O.1  This  inference  accords 
also  with  our  observations  in  connection  with 
the  study  of  saltation  (see  p.  29);  and  those 
observations  suggest  likewise  that  the  curve  is 
materially  modified  by  the  resistances  to  the 
current  involved  in  the  work  of  saltation. 

It  thus  appears  that  in  the  region  with  which 
traction  studies  are  specially  concerned  the 
range  of  Vp  is  great.  The  work  of  traction 
depends  on  a  system  of  velocities  and  nob  on  a 
single  one,  and  there  is  no  individual  value  of 
Vp  with  special  claim  to  the  title  "bed  veloc- 
ity." It  would  be  possible  to  define  bed 
velocity  as  the  value  of  Vp  at  some  particular 
distance  from  the  bed  or  at  a  distance  consti- 
tuting some  particular  fraction  of  the  depth  of 
current;  but  such  a  definition  would  be  hard  to 
apply. 

However  smooth  a  stream  bed  of  debris  may 
be  in  its  general  aspect,  it  is  never  smooth  as 
regards  details.  Figure  53  gives  an  ideal  pro- 


FIGURE  53.—  Ideal  profile  of  a  stream  bed  composed  of  debris  grains. 

tile,  the  intersection  of  a  bed  by  a  vertical  plane. 
Not  only  are  there  salients  and  reentrants,  but 
some  of  the  reentrants  communicate  with  the 
voids  within  the  mass  of  dfibris.  In  many  of 

>  See  Lamb's  Hydrodynamics,  3d  ed.,  p.  544,  1906. 

'  See  Cunningham,  -Allan,  Hydraulic  experiments  at  Roorkee,  p.  46, 
1S75,  and  Inst.  Civil  Eng.  Proc.,  vol.  71,  p.  23,  1882,  where  he  discusses 
t  he  horizontal  velocity  curve;  and  Von  Wagner,  idem,  p.  90. 

155 


156 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


the  reentrants  are  doubtless  stationary  eddies, 
with  reversed  currents  where  the  value  of  Vp  is 
negative.  It  appears  equally  difficult  to  give 
definition  to  the  bed  as  a  datum  from  which  to 
measure  upward,  and  to  select  and  define  a 
locus  for  bed  velocity.  There  is  reason  to  sus- 
pect also  that  the  problem  as  thus  stated  is 
unduly  simplified  by  the  assumption  that  the 
bed  is  a  stable  entity,  clearly  separate  from  the 
zone  of  saltation  above.  It  did  indeed  so  ap- 
pear when  the  process  of  saltation  was  studied 
through  the  glass  wall  of  the  observation 
trough,  but  what  was  witnessed  was  the  phase 
of  the  process  at  the  edge  of  the  channel  bed, 
where  the  current  was  retarded  by  the  resist- 
ance of  the  channel  wall.  At  a  distance  from 
that  wall,  in  the  region  where  the  cloud  of  sal- 
tatory particles  effectually  precludes  visual  ob- 
servation, the  passage  from  stability  to  mobil- 
ity may  be  less  definite.  I  am  led  to  this  sug- 
gestion by  the  observations,  quoted  byMcMath,1 
of  a  civil  engineer  who  descended  in  a  diving 
bell  to  the  bottom  of  the  Mississippi  at  a  point 
where  the  depth  was  65  feet  and  the  bottom  of 
sand.  Stepping  to  the  bed,  he  sank  into  it 
about  3  feet,  and  then  thrusting  his  arm  into  the 
yielding  mass,  could  feel  its  flowing  motion  to  a 
depth  of  2  feet,  the  velocity  diminished  down- 
ward. In  interpreting  these  phenomena,  allow- 
ance must  be  made  for  the  fact  that  the  pres- 
ence of  the  diving  bell  created  an  abnormal 
condition  and  if  it  rested  on  the  bed  put  a  stop 
to  saltation.  The  flow  of  the  sand  is  then  to  be 
ascribed  to  the  difference  in  water  pressure  on 
the  two  sides  of  the  bell.  But  the  fact  of  the 
flow  seems  to  indicate  an  antecedent  state  of 
mobility,  a  laj'-cr  of  the  bed  being  supersatu- 
rated so  as  to  have  the  properties  of  quicksand. 
If  such  a  layer  exists,  then  the  transition  from 
the  bed  to  the  saltation  zone  is  not  abrupt  but 
gradual. 

The  difficulties  in  attempting  to  define  bed 
velocity  are  supplemented  by  those  which  affect 
the  measurement  of  velocities  near  the  bed 
while  traction  is  in  progress  (p.  26),  and  to- 
gether they  have  served  to  prevent  the  use  of 
bed  velocity  as  a  factor  for  quantitative  com- 
parison with  capacity.  This  result  has  been 
regretted  because  the  forces  which  accomplish 
traction  are  applied  directly  through  the  veloc- 
ities of  water  near  the  bed,  and  it  was  admitted 

i  McMath,  R.  E.,  Van  Nostrand's  Mag.,  vol.  20,  p.  227, 1879. 


only  after  the  failure  of  repeated  attempts  to 
obtain  serviceable  estimates  of  bed  velocity. 

In  the  present  chapter  observed  or  interpo- 
lated capacities  are  compared  with  mean  veloc- 
ities of  the  stream,  mean  velocity  being  com- 
puted as  the  quotient  of  measured  discharge  by 
measured  sectional  area.  The  measurements  of 
discharge  and  width  being  relatively  simple  and 
accurate,  the  determinations  of  mean  velocity 
have  the  same  degree  of  precision  as  the  meas- 
urements of  depth.  (See  p.  26.) 

In  comparing  capacity  with  mean  velocity, 
it  is  convenient  always  to  treat  fineness  of 
debris  and  width  of  channel  as  constants,  but 
it  is  also  advantageous  to  recognize  three 
separate  points  of  view  as  to  the  status  of 
discharge,  slope,  and  depth. 

First,  We  may  treat  discharge  as  constant, 
in  which  case  slope  and  depth  vary,  along  with 
velocity  and  capacity.  Each  of  the  observa- 
tional series  (Tables  4,  12,  and  14)  conforms 
to  this  viewpoint.  When  discharge  is  con- 
stant, the  increase  of  power  necessary  to 
increase  velocity  is  given  by  increase  of  slope, 
and  the  increase  of  velocity  causes  the  un- 
changed discharge  to  occupy  less  space.  As 
velocity  and  capacity  increase,  slope  increases 
and  depth  decreases. 

Second,  we  may  treat  slope  as  constant. 
With  slope  constant,  the  increase  of  power 
necessary  to  increase  velocity  is  given  by 
increase  of  discharge,  but  the  rate  at  which 
discharge  is  increased  is  greater  than  the  rate 
of  increase  given  to  velocity,  and  the  increased 
discharge  therefore  requires  more  space.  As 
velocity  and  capacity  increase,  both  discharge 
and  depth  also  increase. 

Third,  we  may  treat  depth  as  constant. 
To  increase  velocity  by  increasing  slope  will, 
as  we  have  seen,  reduce  depth.  To  increase 
velocity  by  increasing  discharge  will,  as  we 
have  seen,  increase  depth.  To  increase  velocity 
without  changing  depth,  it  is  necessary  to 
enlarge  both  slope  and  discharge.  No  experi- 
ments were  conducted  with  fixed  depths,  but 
the  data  for  this  comparison  are  readily 
obtained  by  interpolation. 

It  is  proposed  to  examine  the  relation  of 
capacity  to  velocity  from  each  of  these  view- 
points, developing  the  results  so  far  as  neces- 
sary to  give  a  basis  for  a  comparison  of  the 
viewpoints. 


RELATION   OF   CAPACITY   TO   VELOCITY. 


157 


A  preliminary  remark  applies  to  all.  For 
each4grade  of  d6bris  and  width  of  channel, 
and  for  each  specific  assumption  of  a  constant 
discharge,  slope,  or  depth,  there  is  necessarily 
a  competent  mean  velocity,  below  which  no 
traction  takes  place.  The  conception  of  such 
a  competent  velocity  has  underlain  all  the 
discussions  of  competent  slope,  competent  dis- 
charge, competent  fineness,  and  competent 
form  ratio.  A  broad  analogy  therefore  points 
to  the  propriety  of  formulating  the  capacity- 
velocity  relation  as  other  relations  of  capacity 
have  been  formulated.  And  the  inference 
from  analogy  finds  support  in  logarithmic 
plots  of  C=f(  Vm)  under  each  of  the  three  above- 
mentioned  conditions.  It  may  fairly  be  as- 
sumed, therefore,  that  the  index  of  relative 
variation  for  capacity  and  velocity  itself  varies 
with  velocity,  being  relatively  small  for  high 
velocities,  being  relatively  large  for  low  veloci- 
ties, and  becoming  indefinitely  large  as  com- 
petent velocity  is  approached. 

For  the  purposes  of  this  chapter,  however, 
it  has  seemed  best  to  employ  a  simpler  method, 
using  only  the  synthetic  index  of  relative 
variation — characterized  by  the  symbol  I. 
Calling  the  synthetic  index  for  the  variation 
of  capacity  with  respect  to  mean  velocity  Iv, 
we  may  conveniently  distinguish  by  Ir<t,  Ira 
and  Ivd  the  values  associated  severally  with 
the  special  cases  of  constant  discharge,  con- 
stant slope,  and  constant  depth. 

The  computations  of  the  index  are  made 
chiefly  \>j  the  formula 


..(80) 


log  <7- log  (7" 
logFm'-logTV 


in  which  C'  and  C"  are  specific  capacities,  and 
Vm'  and  Vm"  are  the  corresponding  mean 
velocities.  Graphically,  Iv  is  the  inclination  of 
a  line  connecting  two  points  of  which  the  coor- 
dinates are,  for  the  first,  log  C'  and  log  Vm',  and 
for  the  second,  log  C"  and  log  Vm".  Where 
the  available  data  serve  to  place  more  than  two 
points  on  the  logarithmic  plot  of  C=f(  Vm),  defi- 
nite suggestion  may  thereby  be  made  that  the 
line  connecting  the  extreme  points  does  not 
constitute  the  most  probable  location  of  the 
chord  theoretically  corresponding  to  IY',  and  in 
such  cases  a  line  is  drawn  with  regard  to  all 
the  data,  and  its  inclination  is  measured  on  the 
plot. 


The  subject  of  competent  velocity,  which  is 
of  interest  independently  of  the  formulation  of 
capacity  and  velocity,  will  be  considered  at  the 
end  of  the  chapter. 

THE  SYNTHETIC  INDEX  WHEN  DISCHARGE  IS 
CONSTANT. 

In  Table  14  are  73  series  of  values  of  Vm, 
each  value  corresponding  to  a  stated  value  of  S. 
The  coordinate  series  in  Table  12  contain  values 
of  C  corresponding  to  the  same  values  of  S. 
From  each  pair  of  series  were  taken  the  highest 
and  lowest  values  of  Vm  and  the  corresponding 
values  of  C,  and  from  these  four  quantities  was 
computed  a  value  of  Iv<t.  The  73  values  of  the 
index  are  shown  in  Table  46,  where  the  arrange- 
ment is  such  as  to  exhibit  the  variation  of  the 
values  with  respect  to  discharge. 

TABLE  46. —  Values  of  lyq,  the  synthetic  index  of  relative 
variation  for  capacity  in  relation  to  mean  velocity,  when 
discharge  is  constant. 


Grade. 

w 

Value  of  IfQ  when  discharge  (in  ft.«/sec.)  is— 

0.093 

0.182 

0.363 

0.545 

0.734 

1.119 

(A) 
(B) 

(C) 

(D) 

(E) 
(F) 

(G) 

(H) 

1.32 
1.96 

.23 
.44 
.66 
1.00 
1.32 
1.96 

.44 
.66 
1.00 
1.32 
1.96 

.66 
1.00 
1.32 

.66 
1.00 

.66 
1.00 
1.32 

.66 
1.00 
1.32 

.66 





4.05 
3.55 



3.62 
3.22 

"i'ss" 

5.45 
4.32 
5.76 

4.50 
4.45 
4.61 
5.68 
2.92 

2.93 
5.37 

4.22 

"2."  is" 

3.43 

5.15 
2.51 
3.38 

4.59 

3.49 

6.31 
3.36 

4.30 
3.29 
3.63 
2.54 

2.82 
3.56 
3.90 
4.36 

3.04 
2.52 
3.27 
3.74 

4.59 

2.92 
3.11 
3.69 
3.69 

3.89 

5.38 

4.30 
4.16 

4.13 
5.34 

5.98 
5.35 

6.67 
9.63 

S.  11 

9.22 

4.34 

3.94 
3.34 

6.00 
3.43 

-'•'•'•'•'•'- 

5.10 

7.50 
11.46 







8.30 
8.49 
10.67 

8.97 

10.60 
8.05 
8.26 

8.49 

8.80 
13.60 

12.57 

The  values  which  lie  in  any  horizontal  lino 
agree  as  to  all  conditions  except  discharge.  On 
comparing  the  columns  for  discharges  0.093  and 
0.182  ft.3/sec.,  it  is  seen  there  are  six  lines 
carrying  values  in  both  columns.  The  means 
of  these  values  are  4.80  and  4.21,  the  greater 
mean  belonging  with  the  smaller  discharge.  In 
the  column  for  discharge  0.182  are  nine  values 
coordinate  with  values  hi  the  column  for  dis- 
charge 0.363,  and  the  means  for  the  two  groups 


158 


TRANSPORTATION    OF    DEBRIS   BY   RUNNING    WATER. 


of  nine  are  5.31  and  5.04.  Again  the  greater 
mean  is  associated  with  the  smaller  discharge, 
and  the  same  relation  is  found  by  other  com- 
binations. The  partial  means  resulting  from 
these  reductions  are  arranged  in  the  upper  divi- 
sion of  Table  47.  The  general  fact  exhibited  is 
that  the  index,  other  conditions  being  the  same, 
varies  inversely  with  discharge. 

The  same  method  was  employed  to  discover 
the  nature  of  the  control  of  the  index,  first,  by 
grade  of  d6bris  and,  second,  by  width  of  chan- 
nel; and  the  partial  means  are  grouped  in  the 
middle  and  lower  divisions  of  Table  47.  Of 
the  eight  pairs  of  means  connected  with  grade, 
seven  agree  in  testifying  that  the  index  varies 
inversely  with  fineness.  The  exceptional  testi- 
mony comes  from  the  comparison  of  grades  (B) 
and  (C) ;  and  although  it  is  emphatic,  it  serves 
rather  to  illustrate  the  general  discordance  of 
data  from  the  experiments  with  grade  (B) 
than  to  qualify  the  general  law  as  to  the  index. 

TABLE  47. — Partial  means  based  on  Table  46,  illustrating 
the  control  of  lyq,  by  discharge,  fineness,  and  width. 


Number 
of 

values. 

Means  of  Iv<i  for  discharge  (ft.»/sec.)  of— 

0.093 

0.182 

0.363 

0.545 

0.734 

1.119 

6 
9 

7 

8 

4 
6 
9 
5 
5 
4 
3 
3 

2 
4 
15 
12 

7 

4.80 

4.21 
5.31 

5.04 
3.90 
6.58 



3.51 

""&'.  47' 
5.81 

""5."  64" 

Means  of  7F<Z  for  grade— 

(A) 

(B)      (C) 

(D) 

(K)      (F)       (C) 

(H) 

3.45 

8. 
4. 

49 

01     3.79 
3.34 

4.39 

•••"•  

..    3.21 
3.44 

5.17    .. 
8.73    
5.  19    ..       .     8.71 



8.24     10.54 

9.39 

10.01 

Means  of  Ivl)  for  width  (feet)  of— 

0.23 

0.44 

0.66 

1.00 

1.32 

1.96 

4.97 

4.38 
4.84 

3.80 
5.59 

5.S4 

5.66 

5.87 

3.36 

3.62 

Of  the  five  pairs  of  means  connected  with 
width,  three  show  increase  of  the  index  with 
increase  of  width  and  two  give  the  opposite 
indication.  The  contrasts  are  not  strong  in 
any  case,  and  the  nature  of  the  law  is  not  clear. 


It  may  be  that  the  normal  variation  with 
width  is  so  slight  as  to  be  masked  by  accidental 
errors  of  the  data;  or  it  may  be  that  the  index, 
like  the  constant  a,  is  a  minimum  function  of 
form  ratio. 

By  dividing  the  slope  interval  covered  by  a 
computed  value  of  the  index  and  computing 
separately  the  indexes  for  the  two  subintervals, 
it  was  found  that  the  index  associated  with  the 
higher  slopes  has  a  smaller  value  than  that  for 
the  lower  slopes — that  is,  the  index  varies 
inversely  with  slopes. 

To  compare  the  control  of  capacity  by  mean 
velocity  with  its  control  by  slope  the  73  values 
of  Iw  (the  synthetic  index  of  relative  variation 
of  capacity  and  slope,  under  condition  of  con- 
stant width)  corresponding  to  the  tabulated 
values  of  Ir<t  were  computed.  Each  value  of 
Iv<t  is  greater  than  the  corresponding  value  of 
/„,  the  ratio  ranging  from  1.4  to  4.0.  The 
mean  of  73  values  of  Iv<i  is  5.33;  the  mean  Iw 
is  2.05;  and  the  ratio  of  the  means  is  2.60.  It 
is  in  general  true  that  the  greater  the  indexes 
the  greater  the  ratio  between  them. 

In  Table  48  means  of  the  two  indexes  are 
shown  for  the  several  grades  of  debris.  Each 
index  varies  with  fineness  and  so,  too,  does  the 
ratio  of  indexes. 

TABLE  48.— Synthetic  indexes,  comparing  the  control  of  ca- 
pacity by  -mean  velocity  loith  the  control  by  slope,  and  com- 
paring both  controls  with  grades  of  debris. 


Number 

Grade. 

of 

separate 
determina- 

Mean Ir<l 

Mean  Iw 

Mean  IVQ 

Mean  lw 

Mean  Iw 

Mean  IVq 

tions. 

w 

5 

3.62 

1.87 

1.93 

0.52 

(B) 

18 

4.35 

1.94 

2.24 

.45 

fO! 

19 

3.57 

1.85 

1.93 

.52 

m 

9 

4.50 

1.87 

2.41 

.42 

5 

5.17 

1.83 

2.83 

.35 

F) 

5 

8.73 

2.27 

3.84 

.26 

Q\ 

9 

9.56 

2.60 

3.68 

.27 

H) 

3 

10.01 

3.20 

3.12 

.32 

73 

5.33 

2.05 

2.60 

.38 

MEAN  VELOCITY  VERSUS  SLOPE. 

The  comparison  of  IV(t  with  Iw  affords, 
incidentally,  an  estimate  of  the  relative  varia- 
tion of  mean  velocity  and  slope.  The  rate  of 
variation  of  capacity  with  mean  velocity  being 
IVQ,  the  rate  of  variation  of  mean  velocity  with 
capacity  is  I//F«;  and  the  rate  of  variation  of 
capacity  with  slope  being  Iw,  the  rate  of 
variation  of  mean  velocity  with  slope  is 


RELATION   OP   CAPACITY   TO   VELOCITY. 


159 


/w,//v«.  The  values  of  Iw/Iv<t  listed 
in  Table  48  are  therefore  estimates  of  the  rela- 
tive variation  of  mean  velocity  in  relation  to 
slope,  and  they  have  the  same  quality  as  the 
corresponding  values  of  /„,  and  JVQ.  They  are 
subject  to  the  limiting  conditions  of  constant 
discharge,  fineness,  and  width,  and  they  are  av- 
erages of  variability  for  practically  the  entire 
range  of  conditions  realized  in  the  laboratory, 
with  the  exception  of  those  in  the  immediate 
neighborhood  of  competence.  For  this  range 
of  conditions  mean  velocity  varies,  on  the 
average,  as  the  0.38  power  of  the  slope. 

This  result  is  comparable  with  the  generali- 
zation embodied  in  the  Chezy  formula,  which 
makes  mean  velocity  vary  with  the  0.5  power 
of  the  slope.  The  two  are  not  inconsistent 
because  they  pertain  severally  to  loaded  and 
loadless  streams.  In  a  loaded  stream  increase 
of  slope  augments  load  and  thus  develops 
rapidly  a  factor  of  resistance  from  which  the 
loadless  stream  is  free.  Velocity,  being  lim- 
ited by  resistances,  develops  less  rapidly  when 
the  conditions  are  such  that  the  resistances 
develop  more  rapidly. 

THE  SYNTHETIC  INDEX  WHEN  SLOPE  IS 
CONSTANT. 

For  a  selected  slope,  values  of  Vm  may  be 
found  in  Table  14  which  agree  as  to  width  of 
channel  and  grade  of  debris  and  differ  only  as 
to  the  discharges  with  which  they  are  asso- 
ciated; and  in  Table  12  may  be  found  the  cor- 
responding values  of  capacity.  Such  pairs  of 
values,  when  occurring  in  series  of  two  to  five, 
constitute  data  for  the  computation  of  values 
of  Iva.  To  cover  the  entire  range  of  tabulated 
data  without  needless  repetition  choice  was 
made  of  slopes  0.6,  1.0,  1.4,  and  2.0  per  cent. 
The  data  associated  with  these  slopes  gave  66 
values  of  the  index,  the  values  being  essen- 
tially independent  except  in  a  few  instances. 
They  are  shown  in  Table  49.  From  them  were 
derived  the  sets  of  partial  means  arranged  in 
Table  50,  the  method  of  derivation  being  that 
described  in  connection  with  Table  47. 

The  means  show  that  Iva  varies  inversely 
with  slope,  the  variation  being  of  moderate 
amount.  They  leave  little  question  that  it 
varies  inversely  with  fineness,  though  the  evi- 
dence is  somewhat  conflicting.  They  indicate 


also  a  direct  variation  with  trough  width,  the 
opposite  tendency  being  indicated  only  by  the 
means  for  the  greatest  widths.  The  plots  of 
the  data,  not  here  reproduced,  show  that  the 
index  varies  inversely  with  discharge. 

TABLE  49.—  Values  of  lys,  the  synthetic  index  of  relative 
variation  for  capacity  in  relation  to  mean  velocity,  when 
slope  is  constant. 


Grade. 

S 

Value  of  Iyg  for  width  (feet)  of  — 

0.23 

0.44 

0.66 

1.00 

1.32 

1.96 

(A) 
(B) 

(C) 

(D) 
(E) 

(F) 

.06 

.6 
1.0 
1.4 
2.0 

.6 
1.0 
1.4 
2.0 

.6 
1.0 
2.0 

.6 
1.0 
1.4 
2.0 

1.0 
1.4 
2.0 

1.0 
1.4 
2.0 

1.4 
2.0 

2.18 

3.24 
3.40 
3.24 

4.20 
3.48 
3.10 

1.87 

5.15 
4.62 

3.82 
4.39 
3.80 
4.95 

3.20 

2.98 
2.34 
1  59 

4.60 
4.10 
4.45 
4.17 

3.31 
3.01 
3.35 

2.42 
3.30 

2.60 

4.38 
2.83 
2.89 

2.80 
2.45 
2.03 
1.33 

3.55 
3.40 
2.72 

2.55 
2.49 
2.85 

3.38 

4.10 





3.68 
2.29 

3.06 
2.85 
2  79 

2.39 

2.21 

4.86 
4.92 

3.97 
3.63 
3.50 

9  78 

2.94 
5.80 
4.72 

4.62 
4.92 
3.10 

:::;:::: 

6.50 
2.06 
5.05 

5.84 

TABLE  50.— Partial  means  based  on  Table  49,  illustrating 
the  control  of  IYS  by  slope,  fineness,  and  width. 


Number 
of 
values. 

Means  of  Isy  lor  slope  (per  cent)  of— 

0.6 

1.0 

1.4 

2.0 

12 
16 
11 

2 
13 
6 
4 
£ 
2 

1 

4 
18 
11 
6 

3.68 

3.35 
3.63 

3.43 
4.23 

3.81 

Means  of  Ivs  for  grade— 

(A)      (B) 

(C) 

(D) 

(E)       (F) 

(G) 

(H) 

2.02     4.19 
3.81 

2.87 
30.5 

"3.'  13    '. 

2.82 

3.23    ...... 
2.93     4.11 

Tos 

3.56 

7.81 

Means  of  Ivg  for  width  (fee 

t)of— 

0.23          0.44 

0.66 

1.00 

1.32 

1.96 

2.42 

2.60 
3.17 

3.23 
3.57 

3.47 
3.72 
.        3.46 

3.54 

'"i'.n 

3.27 

2.97 

"'3.'  is' 



160 


TRANSPORTATION    OF    DEBRIS   BY    RUNNING    WATER. 


THE   SYNTHETIC   INDEX  WHEN   DEPTH   IS 
CONSTANT. 

The  capacities  and  mean  velocities  corre- 
sponding to  particular  depths  were  derived  from 
the  computation  sheets  described  on  page  95. 
The  selected  depths  were  0.10,  0.14,  0.20,  and 
0.28  foot,  and  the  data  available  for  these  depths 
afforded  32  values  of  the  synthetic  index,  Ivd- 
These  arc  recorded  in  Table  51,  and  partial 
means  derived  from  them  are  arranged  in 
Table  52  so  as  to  show  variation  in  relation  to 
fineness  of  debris  and  width  of  channel. 

TABLE  51. —  Values  of  Ira,  the  synthetic  index  of  relative 
variation  for  capacity  in  relation  to  mean  velocity,  when 
depth  of  current  is  constant. 


Width 
(feet). 

Depth 

(feet). 

Value  of  Ivd  for  grade— 

(A) 

(B) 

(C) 

(D) 

(K) 

(G) 

0.44 
.66 

1.00 
1.32 
1.96 

0.14 

.10 
.14 
.20 

.28 

.14 
.20 

.28 

.10 
.14 
.20 

.10 
.14 
.20 

3.73 

2.59 
3  16 

4  12 

3.51 

3.39 
2.62 

3.59 

2.80 
3.44 



"  ~4.'  02"  ' 

3.17 
3.51 

"4."  46" 



7.86 

3.05 
2.17 
2.03 

3.92 
3.41 
4.50 

3.86 
3.10 

3.23 
3.65 

3.52 

7.22 

3.02 
3.60 
4.27 

2.36 
2.23 

TABLE  52. — Partial  means  based  on  Table  51,  illustrating  the 
control  of  IVt,  by  fineness  and  width. 


Number 
of 

values. 

Means  of  Ivd  for  grade  — 

(A) 

(B) 

(C) 

(D) 

(E) 

(0) 

4 
8 
3 
1 
2 

1 

4 
3 

7 

2.87 

3.03 
3.59 

3.40 
3.16 
3.44 

3.40 

4.46 

3.04 

7.54 

Means  of  Ivd  for  width  (feet)  of  — 

0.44 

0.66 

1.00 

1.32 

1.96 

3.73 

3  28 

3.17 

3.46 
3.71 

2.88 
3.00 

3.09 

With  a  single  exception  in  each  group  of 
means,  the  indication  is  that  the  index  varies  in- 
versely with  fineness  and  directly  with  width. 
Under  the  condition  of  constant  depth  each 
change  in  mean  velocity  13  accompanied  by 
changes  of  both  slope  and  discharge,  and  the 
influences  of  the  two  can  not  be  examined  sepa- 


rately. The  features  of  the  logarithmic  plots, 
not  here  reproduced,  show  that  the  index  varies 
inversely  with  slope  and  discharge,  considered 
together. 

THE  THREE  INDEXES. 

In  bringing  together  the  results  outlined  in 
the  preceding  paragraphs  we  may  replace  width 
of  channel  by  form  ratio,  bearing  in  mind  that 
the  two  factors  are  so  related  that  their  varia- 
tions are  in  opposite  senses.  So  far  as  qualita- 
tive statement  is  concerned,  the  three  syn- 
thetic indexes  are  identical  in  properties.  The 
sensitiveness  of  capacity  for  traction  to  the 
control  of  mean  velocity  of  current  varies  in- 
versely with  slope  of  channel,  discharge,  fine- 
ness of  debris,  and  form  ratio. 


(81) 


As  to  the  first  three  conditions  the  generaliza- 
tion is  unqualified,  but  it  is  possible  that  the 
function  as  to  form  ratio  is  of  the  minimum 
class  instead  of  inverse. 

The  three  indexes  are  not  of  the  same  mag- 
nitude. In  comparing  them  Ira  was  made  the 
standard,  partly  because  the  values  computed 
for  it  are  fewer.  To  compare  with  each  of  its 
32  values,  that  value  of  Irs  which  most  nearly 
represents  the  same  group  of  conditions  was 
selected,  and  also  a  pair  of  values  of  Iv<t  which 
collectively  represent  nearly  the  same  condi- 
tions. Means  were  then  derived  for  each  index 
for  each  of  the  six  grades  to  which  they  pertain, 
and  also  general  means  —  all  of  which  are  shown 
in  Table  53.  In  the  same  table  are  the  ratios 
between  general  means  and  also  between  the 
corresponding  grade  means.  For  the  general 
means  Ir<i  is  7  per  cent  greater  than  IYd,  and  7r« 
is  the  greater  for  all  the  partial  means  but  one. 
For  all  the  partial  means  but  one  IVs  is  smaller 
than  Ivd,  and  for  the  general  means  it  is  9  per 
cent  smaller.  Capacity  is  most  sensitive  to 
the  mean  velocity  conditioned  by  constant  dis- 
charge and  least  sensitive  to  the  mean  velocity 
conditioned  by  constant  slope. 

These  results  have  a  theoretic  connection 
with  the  fact  that  capacity  is  more  sensitive 
to  changes  of  slope  than  to  those  of  discharge. 
When  discharge  is  constant  the  changes  of 
velocity  are  caused  by  changes  of  slope,  and 
the  changes  in  capacity  are  those  due  to  the 
changes  of  slope.  When  slope  is  constant  the 
changes  of  velocity  are  caused  by  changes  of 


RELATION   OF   CAPACITY    TO   VELOCITY. 


161 


discharge,  and  the  changes  in  capacity  are 
those  due  to  changes  of  discharge.  When 
depth  is  constant  the  changes  of  velocity  are 
caused  by  concurrent  changes  of  slope  and 
discharge,  and  the  changes  in  capacity  are 
intermediate  between  those  caused  by  slope 
alone  and  by  discharge  alone. 

TABLE  53. — Comparison  of  synthetic  indexes  of  relative  vari- 
ation for  capacity  and  mean  velocity,  -under  the  several  con- 
ditions of  constant  discharge,  constant  depth,  and  con- 
stunt  slope. 


Means. 

Ratios. 

Number 

Grade. 

of  com- 

parisons. 

IVQ 

Ivd 

IYS 

^ 

hs. 

ht 

'vt 

(A) 

4 

3.34 

2.87 

1.97 

1.16 

0.69 

(B) 

9 

3.55 

3.41 

4.19 

1.04 

1.23 

(C) 

13 

3.60 

3.  55 

2.96 

1.01 

.83 

(D) 

3 

4.14 

3.40 

2.63 

1.22 

.77 

(E) 

1 

4.39 

4.46 

3.06 

.99 

.69 

(G) 

2 

8.84 

7.54 

3.86 

1.17 

.51 

General  means  — 

3.98 

3.  US 

3.21 

1.07 

.91 

It  is  profitable  to  consider  the  same  facts 
also  in  relation  to  depth  of  current.  Postulate 
an  initial  status,  with  a  particular  discharge 
and  slope,  determining  a  certain  velocity, 
depth,  and  capacity.  First,  increase  the  slope 
until  the  velocity  is  doubled.  The  capacity  is 
increased,  let  us  say  (borrowing  mean  Iv<t 
from  the  table)  to  15.8  times  its  initial  amount. 
At  the  same  time  the  depth  is  reduced  one- 
half.  Second,  after  returning  to  the  initial 
status,  increase  slope  and  discharge  by  such 
amounts  as  to  double  the  velocity  without 
changing  the  depth.  The  capacity  grows 
(mean  Ivd)  to  12.8  tunes  its  original  amount. 
Third,  starting  from  the  initial  condition  as 
before,  increase  the  discharge  until  velocity  is 
doubled.  The  capacity  grows  (mean  7K»)  to 
9.2  times  its  original  amount;  and  the  depth 
is  at  the  same  time  increased,  being  more  than 
doubled.  Thus,  for  the  same  (doubled)  mean 
velocity,  the  capacity  is  greater  as  the  depth 
is  smaller.  Mean  velocity  is  more  efficient  for 
traction  as  depth  is  less. 

Now,  the  primary  direct  cause  of  stream 
traction  is  bed  velocity.  A  concurrent  cause 
theoretically  exists  in  the  component  of  gravity 
parallel  to  the  slope,  acting  directly  on  the 
load,  but  for  all  ordinary  stream  slopes  this 
factor  is  negligible.  Slope  and  discharge  are 
(essentially)  indirect  causes  and  are  causes 
only  in  so  far  as  they  occasion  bed  velocity. 
They  also  determine  mean  velocity,  and,  from 


one  point  of  view,  mean  velocity  may  be  said 
to  control  capacity  by  controlling  bed  velocity. 

Let  us  assume,  for  the  moment,  that  bed 
velocity  determines  capacity  irrespectively  of 
depth.  Then  the  variations  of  capacity  above 
described  imply  corresponding  variations  of  bed 
velocity,  and,  as  the  mean  velocity  does  not 
change,  we  may  infer  that  the  ratio  of  bed  ve- 
locity to  mean  velocity  is  a  function — a  de- 
creasing function — of  depth. 

This  proposition  is,  to  say  the  least,  worthy 
of  consideration,  but  it  fails  of  demonstration 
because  the  assumption  which  paved  the  way 
for  it  is  not  valid.  It  is  not  true  that  the  rela- 
tion of  capacity  to  bed  velocity  is  independent 
of  depth.  In  the  first  place,  change  of  depth, 
when  not  accompanied  by  change  of  mean  ve- 
locity, causes  change  in  the  mode  of  traction. 
Within  the  range  of  the  above  hypothetic  con- 
ditions may  occur  both  the  dune  rhythm  and 
the  antidune  rhythm;  and  at  least  one  of  these 
has  an  influence  on  capacity.  Moreover,  these 
rhythms  involve  diversity  of  velocity  from 
point  to  point  along  the  bed,  so  that  "  bed  ve- 
locity" has  not  a  simple  definition. 

In  the  second  place,  the  load,  or  the  work  of 
traction,  reacts  on  the  vertical  distribution  of 
velocities.  In  figure  54  the  line  ABO  is  as- 


FIGCBE  54.— Ideal  curves  of  velocity  in  relation  to  depth,  illustrating 
their  relation  to  the  zone  of  saltation.  0— zero  of  velocity  (horizontal) 
and  of  distance  from  the  bottom  (vertical). 

sumed  as  the  vertical  velocity  curve  of  a  stream 
flowing  in  a  straight  conduit  and  bearing  no 
load.  A'B'C,  identical  except  as  to  vertical 
dimensions,  is  assumed  as  the  curve  correspond- 
ing to  the  same  mean  velocity  in  a  current  one- 
half  as  deep.  Introducing,  now,  the  condition 
of  traction,  we  may  represent  the  upper  limit 
of  the  zone  of  saltation  by  the  line  DE.  The 
potential  velocities  within  the  zone  are  evi- 
dently quite  different  for  the  two  depths  of  cur- 
rent, and  they  give  advantage,  for  traction,  to 
the  shallower  current.  The  work  of  saltation 
tends  to  retard  the  lower  filaments  of  current 
and  through  these  the  higher  filaments,  reduc- 


200218— No.  86—14- 


-11 


162 


TRANSPORTATION   OF   DEBBIS   BY   RUNNING   WATER. 


ing  the  mean  velocity  and  increasing  the  depth. 
If  mean  velocity  and  depth  be  restored  by  suit- 
able changes  in  slope  and  discharge,  a  new  and 
different  vertical  velocity  curve  will  be  obtained 
for  each  depth.  While  I  am  not  able  to  de- 
duce the  exact  nature  of  the  changes,  it  seems 
clear  that  those  portions  of  the  new  curves 
within  the  zone  of  saltation  will  be  contrasted 
in  some  such  way  as  are  the  potential  curves  of 
the  drawing,  and  that  for  purposes  of  traction 
the  advantage  will  still  be  with  the  shallow 
current.  Provided  the  portions  of  the  ve- 
locity curves  within  the  zone  of  saltation  show 
steeper  gradient  for  the  current  of  less  depth, 
tractional  capacity  will  for  that  current  bear 
a  higher  ratio  to  mean  velocity. 

RELATIVE  SENSITIVENESS  TO  CONTROLS. 

The  synthetic  index  of  relative  variation  for 
capacity  in  relation  to  mean  velocity,  when  tho 
limiting  condition  is  constant  discharge,  namely, 
Ira,  has  been  estimated  (p.  158)  as  2.60  times 
the  corresponding  index,  Iw  for  capacity  in 
relation  to  slope.  The  same  index  has  been 
estimated  (Table  53)  as  1.07  times  IVd  and  1.19 
times  Ira.  Combination  of  ratios  indicates 
that  Ivd  is  2.43  times  and  Ira  2.18  times  as 
great  as  /„.  While  these  figures  have  an 
appearance  of  exactitude,  their  order  of  pre- 
cision is  really  low.  They  are  built  on  aver- 
ages of  individual  values  of  indexes,  which 
among  themselves  are  highly  diversified.  At 
best  they  represent  the  average  of  values 
covered  by  the  range  of  experiments  in  the 
laboratory,  but  hi  part  they  are  based  on 
values  covering  much  narrower  ranges.  More- 
over, it  was  not  possible,  except  in  the  case  of 
Ivq  and  /„,  to  derive  the  compared  series  of 
values  of  the  index  from  data  representing 
exactly  the  same  conditions.  For  these  rea- 
sons the  numerical  results  should  be  accepted 
only  as  indicating  an  order  of  sequence  and  an 
order  of  magnitude.  The  quantitative  re- 
sponse of  capacity  to  the  change  of  mean 
velocity  is  much  larger  than  its  response  to 
change  of  slope,  probably  more  than  twice  as 
large.  Minor  differences  depend  on  the  con- 
ditions under  which  mean  velocity  varies. 
The  response  is  greatest  when  velocities  are 
subject  to  the  condition  of  constant  discharge, 
less  when  the  restrictive  condition  is  constant 
depth,  and  least  when  it  is  constant  slope. 


COMPETENT  VELOCITY. 

The  demonstrations  by  Leslie,  Hopkins, 
Airy,  and  Law  of  the  proposition  that  the 
diameter  of  the  largest  particle  a  current  can 
move  is  proportional  to  the  square  of  the 
velocity  involve  the  principle  that  the  pressure 
of  a  current  is  proportional  to  the  square  of  its 
velocity,  and  also  the  assumption  that  the 
forward  pressures  on  different  parts  of  the 
particle  are  the  same,  so  that  the  total  pressure 
is  proportional  to  the  sectional  area  of  the 
particle.1  Under  that  assumption  the  total 
pressure  may  be  conceived  as  applied  to  the 
center  of  gravity  of  the  particle,  a  considera- 
tion of  importance  when  the  motion  given  to 
the  particle  is  of  the  nature  of  rolling  or  over- 
turning. These  assumptions  are  not  strictly 
true,  because  in  the  immediate  vicinity  of  the 
channel  bed  the  velocity  increases  with  dis- 
tance from  the  bed.  Moreover,  as  wo  have 
seen  (p.  29),  the  rate  of  increase  is  a  diminish- 
ing rate.  As  a  consequence  of  the  inequality 
of  velocity  and  its  mode  of  distribution  (1) 
the  average  pressure  on  the  upstream  face  of  a 
large  particle  is  greater  than  the  corresponding 
average  pressure  on  a  small  particle,  (2)  the 
point  of  application  of  the  total  pressure  (the 
point  which  determines  the  lever  arm  in  over- 
turning and  rolling)  is  always  above  the  center 
of  gravity,  and  (3)  the  point  of  application  may 
be  differently  related  to  the  center  of  gravity 
in  particles  of  different  sizes.  The  general 
effect  of  these  qualifying  circumstances  is  to 
reduce  the  difficulty  of  moving  large  particles, 
and  thus  to  make  the  rate  at  which  competent 
diametar  of  particle  increases  with  velocity  (at 
any  particular  level)  somewhat  greater  than 
that  of  the  square  of  the  velocity. 

On  the  other  hand,  it  is  to  be  observed  that 
in  stream  traction  the  roughness  of  the  channel 
bed  is  denned  by  the  coarseness  of  the  load, 
and  the  system  of  velocities  near  the  bed  is  a 
function  of  several  things,  one  of  which  is  the 
roughness.  It  is  by  no  means  impossible  that 
the  vertical  velocity  curve  of  a  stream  flowing 
over  a  bed  of  coarse  debris  is  an  enlarged 
replica  of  the  curve  of  a  shallower  stream 
flowing  over  a  bed  of  finer  debris,  in  which  case 
the  law  of  Leslie  might  hold  despite  the  quali- 
fications mentioned  above.  The  problem  is  too 

1  For  references  see  footnote  on  page  16. 


RELATION   OF   CAPACITY   TO   VELOCITY. 


163 


complicated,  for  the  present  at  least,  for  full 
deductive  treatment,  and  there  are  no  adequate 
experimental  data. 

Most  of  the  earlier  experiments  on  compe- 
tence pertained  to  flume  traction,  but  it  is 
probable  that  in  those  of  T.  Login  (1857)  the 
conditions  were  such  as  to  give  stream  traction. 
The  currents  he  employed  were  shallow,  and  he 
measured  velocities  by  means  of  floats  which 
occupied  half  the  depth  of  the  water.1  As 
dimensions  of  transported  particles  are  not 
included  in  his  report  of  observations,  a  numer- 
ical formula  can  not  be  based  on  it. 

Experimental  data  on  competent  velocity  for  stream  traction. 

[By  T.  Login.] 

Velocity 
in  ft.  /sec. 

Brick  clay,  mixed  with  water  and  then  allowed  to 
settle  .........................................     0.  25 

Fresh-water  sand  ..................................  67 

Sea  sand  .........................................     1.10 

Rounded  pebbles,  size  of  peas  ....................     2.  00 

John  S.  Owens,  experimenting  on  the  trans- 
porting power  of  sea  currents,  made  use  of 
small  streams  flowing  from  one  tide  pool  to 
another.  Measuring  velocities  by  means  of 
floats,  he  tested  the  ability  of  currents  to  move 
pebbles,  0.5  inch  to  6  inches  in  diameter,  over 
a  channel  bed  of  sand.2  His  results  are  formu- 
lated in 

45 


where  Dl  is  diameter  of  pebble  in  inches,  V 
velocity  in  ft./sec.,  and  TPthe  weight  in  pounds 
of  a  cubic  foot  of  the  material  of  the  pebble. 
This  is  equivalent  to 


.(82) 


0.059 
J'G-1   ' 


where  D  is  the  diameter  in  feet  and  G  the  den- 
sity of  the  material.3 

Of  the  Berkeley  experiments  on  compe- 
tence, recorded  in  Tables  9  and  10,  a  single 
series  bears  on  the  point  under  consideration, 
but  its  bearing  is  less  diiect  than  could  bo 

'  Royal  Soc.  PMinburgh  Proc.,  vol.  3,  p.  475,  1857. 

2  Geog.  Jour.,  vol.  31,  pp.  415-420,  1908. 

»  See  also  experiments  by  T.  E.  Blackwell,  cited  in  Chapter  XII. 


desired.  By  selecting  from  Table  9  data  cor- 
responding to  a  trough  width  of  1  foot  and  a 
discharge  of  0.363  ft.3/sec.  I  was  able  to  make  a 
logarithmic  plot  of  diameters  of  particles  of 
grades  (B),  (C),  (D),  (E),  (G)  in  relation  to 
competent  mean  velocities,  and  this  plot  gave 
the  following  equation: 


£-0.0025 


Fm"....(83) 


The  exponent  2.7  is  connected  with  the  fact 
that  Vm  in  this  case  is  mean  velocity  with  dis- 
charge constant.  It  is  easy  to  infer  from  the 
data  assembled  in  Table  53  that  if  either  depth 
or  slope  had  been  the  constant  condition  in  the 
experiments  a  smaller  exponent  would  be  in- 
dicated. So  the  possibility  remains  that  the 
Leslie  law  is  true  of  mean  velocities  provided 
the  depths  increase  along  with  the  velocities, 
and  the  experimental  data  manifestly  do  not 
apply  to  bed  velocities. 

In  order  to  compare  the  Berkeley  observa- 
tions with  Login's,  the  diametera  of  his  mate- 
rials have  been  computed  by  equation  (83). 
The  diameter  found  for  brick  clay,  0.00006  foot, 
is  much  too  large;  but  this  result  is  readily 
accounted  for  by  the  fact  that  adhesion  is  an 
important  factor  in  the  resistance  of  clay  to 
the  action  of  the  current.  If  the  computed 
diameters  for  his  "fresh-water  sand,"  0.00085 
foot,  and  "sea  sand,"  0.0032  foot,  are  correct, 
those  materials  correspond  severally  to  our 
grades  (A)  and  (D),  a  very  fine  sand  and  a 
coarse  sand.  The  diameter  found  for  "peb- 
bles, size  of  peas,"  is  0.016  foot,  or  one-fifth  of 
an  inch. 

The  coefficient  obtained  by  Owens,  0.059  in 
equation  (82),  is  14  times  as  large  as  our  coeffi- 
cient, 0.0042  in  equation  (83),  a  contrast 
which  accords  in  a  general  way  with  the  differ- 
ence between  the  classes  of  phenomena  ob- 
served. The  pebbles  he  tested  were  rolled 
over  a  bed  of  relatively  fine  material,  which 
gave  them  a  smooth  pathway  with  little  resist- 
ance, while  the  grains  to  be  moved  in  our 
experiments  rested  among  similar  grains  and 
were  less  readily  dislodged. 


CHAPTER  VIIL— RETATION  OF  CAPACITY  TO  DEPTH. 


INTRODUCTION. 

As  a  condition  controlling  capacity  for  trac- 
tion, depth  has  several  distinct  aspects;  and  the 
nature  of  its  control  depends  altogether  on  the 
character  of  associated  conditions.  Three  as- 
pects will  here  be  considered.  They  all  assume 
that  size  of  debris  and  width  of  channel  are 
constant,  and  they  are  severally  characterized 
by  the  limiting  conditions  of  constant  discharge, 
constant  slope,  and  constant  mean  velocity. 
In  examining  the  nature  of  the  controls,  and  in 
comparing  them  with  one  another  and  with 
other  controls,  use  will  be  made  of  the  syn- 
thetic index;  and  the  method  of  discussion  will 
be  similar  to  that  of  the  preceding  chapter. 
The  symbols  for  the  synthetic  index,  under  the 
three  limiting  conditions,  will  be  severally 

Ida,  Ids,  and  Idr. 
WHEN  DISCHARGE  IS  CONSTANT. 

A  stream  of  constant  discharge,  flowing  in  a 
channel  of  constant  width,  can  change  its  depth 
only  by  changing  its  mean  velocity,  and  depth 
and  velocity  vary  in  opposite  senses.  What- 
ever the  ratio  by  which  the  mean  velocity  is  in- 
creased or  diminished,  the  depth  diminishes  or 
increases  in  the  same  ratio.  It  follows  that 
the  law  of  change  for  capacity  in  relation  to 
mean  velocity  is  the  inverse  of  the  law  of  change 
for  capacity  in  relation  to  depth.  So  far  as 
these  laws  are  expressed  by  values  of  the  syn- 
thetic indexes, 

*d«  =  ~  IVQ- 

Independent  computations  of  IdQ  are  there- 
fore unnecessary,  as  the  values  of  Irq  in  Tables 
46,  47,  and  48  need  only  change  of  sign  to 
become  the  corresponding  values  of  7d«.  The 
following  summary  statement  of  the  general 
features  of  the  control  of  capacity  by  depth 
is  but  a  condensation  and  adaptation  of  the 
statement  on  pages  157-158. 

Under  the  condition  of  constant  discharge 

capacity  varies  inversely  with  depth.     Its  rate 

of  variation  is  more  than  twice  the  rate  at 

which  it  varies  with  slope   of  channel.     The 

164 


rate  responds  to  changes  in  discharge,  slope  of 
channel,  fineness  of  debris,  and  form  ratio, 
diminishing  as  those  factors  increase  (with 
possible  exception  as  to  form  ratio). 

Depth,  when  subject  to  the  condition  of  con- 
stant discharge,  varies  inversely  with  slope. 
Estimates  of  the  average  rate  of  variation  are 
contained  in  the  last  column  of  Table  48. 

WHEN  SLOPE  IS  CONSTANT. 

A  stream  flowing  down  a  constant  slope,  in 
a  channel  of  constant  width,  and  transporting 
debris  of  a  particular  grade  changes  its  depth 
when  the  discharge  is  changed.  The  depth  is 
greater  as  the  discharge  is  greater.  The 
capacity  for  traction  also  is  greater  as  the 
discharge  is  greater.  Therefore  the  capacity 
varies  in  the  same  sense  as  the  depth. 

Table  54  contains  68  values  of  Ids,  com- 
puted from  data  of  Tables  12  and  14.  The 
method  of  derivation  was  identical  with  that 
already  described  for  Table  49,  with  the  excep- 
tion that  values  of  depth  were  used  instead  of 
values  of  mean  velocity. 

TABLE  54. —  Values  of  Ids,  the  synthetic  index  of  relative  vari- 
ation for  capacity  in  relation  to  depth  of  current,  when  slope 
is  constant. 


Grade. 

S 

Value  of  /a*  for  width  (feet)  of— 

0.23 

0.44 

0.66 

1.00 

1.32 

1.% 

(A) 
(B) 

(C) 
(D) 

(E) 

(F) 
(G) 

(H) 

O.fi 

.6 
1.0 
1.4 

2.0 

.6 
1.0 
1.4 
2.0 

.6 
1.0 
1.4 
2.0 

.6 
1.0 
1.4 
2.0 

1.4 
2.0 

1.0 
1.4 
2.0 

1.0 
1.4 
2.0 

2.31 

2  93 
2.  52 
2.36 

2.28 

2.14 
1.85 
1.41 

2.02 
2.05 
1.78 
1.59 

2.27 
1.98 
2.34 
2.27 

1.84 
2.18 
2.01 
1.87 

2.40 
2.18 
1.97 
1.83 

2.51 
2.22 
2.04 
2.22 

1.97 
2.30 
2.16 
2.75 

1.90 

1.07 
.89 

2.17 

3.44 
2.12 
1.80 

2.  5fi 
2.85 
2.36 

3.29 
2.13 
2.65 

2.03 
2.12 

1.02 
1.09 

1.70 
1.79 
2.00 

3.59 

2.87 

3.02 
2.91 
2.51 

2.40 
2.03 

2.74 
2.18 
1.96 

2.37 

3.88 
3.07 
3.01 

2.95 
2.22 

RELATION   OF   CAPACITY   TO   DEPTH. 


165 


The  derivation  of  moans  in  Table  55  followed 
the  precedents  of  Tables  47  and  50.  An  ex- 
amination of  logarithmic  plots  shows  that  7<jS 
varies  inversely,  but  only  slightly,  with  dis- 
charge. The  means  in  Table  55  show  that  it 
varies  inversely  with  slope  and  fineness,  but 
they  give  no  clear  indication  of  its  essential  re- 
lation to  width. 

TABLE  55. — Partial  means  based  on  Table  54,  illustrating 
the  control  of  Ids,  by  slope,  fineness,  and  width. 


Num- 
ber of 
values. 

M 

eans  of  Ids  for  slope  (per  cent)  of  — 

0.6 

1.0 

1.4 

2.0 

14 
18 
13 

2 
15 
10 
6 
4 
8 
3 

2 
4 
19 

11 
6 

2.66 

2.13 
2.12 

2.10 
2.40 

2.24 

Means  of  las  for  grade  — 

(A) 

(B) 

(C) 

(D) 

(E) 

(F) 

(G) 

(H) 

2.29 

2  53 

2.08 
2.13 

2.39 
2.33 

2.12 

2.22 

2.23 

L.55 

2.22 
2.28 

2.20 

2.72 

2.78 

1.81 

2.20 

2.02 

2.29 

2.51 

Means  of  /,jg  for  width  (feet)  of— 

0.23 

0.44 

0.66 

1.00 

1.32 

1.96 

0.98 

1.91 

2.38 

2.16 
1.98 

2.36 
2.37 

2.70 
2.59 

2.24 

For  each  of  the  68  values  of  lds,  the  corre- 
sponding value  of  the  synthetic  index  of  the 
relation  of  capacity  to  discharge,  I3,  was  com- 
puted, the  relations  being  such  that  the  paired 
values  represent  exactly  the  same  conditions. 
It  was  found  that,  without  exception,  the  values 
of  Ids  are  greater  than  the  companion  values 
of  73.  Table  56  contains  a  series  of  compara- 
tive means  and  their  ratios,  the  ratio  of  general 
means  being  1.62. 

TABLE  56. — Synthetic  indexes,  comparing  the  control  of  ca- 
pacity by  depth  with  the  control  by  discharge,  and  comparing 
ooth  controls  with  grades  of  debris. 


Grade. 

Number 
of 
separate 
determi- 
nations. 

Mean  Ids. 

Mean  /3. 

Mean  las 

Mean/3 

Mean  It 

Maan/ds 

(A 

((CB) 

$ 

* 

In 

2 
17 
17 
10 
6 
4 
9 
3 

2.29 
1.95 
2.41 
2.12 
1.58 
2.72 
2.81 
2.51 

1.37 
.30 
.36 
.26 
.99 
.81 
.76 
.70 

.67 
.50 
.77 
.68 
.59 
1.50 
1.59 
1.48 

0.60 
.67 
.56 
.59 
.63 
.67 
.63 
.68 

68 

2.25 

1.39 

1.62 

.  r,2 

DEPTH  VERSUS  DISCHARGE. 

The  comparison  of  7ds  with  73  affords,  inci- 
dentally, an  estimate  of  the  relative  variation 
of  depth  and  discharge.  The  rate  of  variation 
of  capacity  with  depth  being  Ids,  the  rate  of 
variation  of  depth  with  capacity  is  1/7^;  and, 
the  rate  of  variation  of  capacity  with  discharge 
being  73,  the  rate  of  variation  of  depth  with  dis- 
charge is  1 1 ha  +  73  =  73/7ds.  The  values  of  I3/Ids 
in  Table  56  are  therefore  estimates  of  the  varia- 
tion of  depth  in  relation  to  discharge,  under  the 
limiting  condition  of  constant  slope.  They  are 
of  the  quality  of  the  synthetic  index  and  are 
based  on  the  general  range  of  conditions 
realized  in  the  laboratory,  except  those  in  the 
neighborhood  of  competence.  For  this  range 
of  conditions  depth  varies,  on  the  average,  with 
the  0.62  power  of  discharge. 

WHEN  VELOCITY  IS  CONSTANT. 

By  interpolation  from  the  data  recorded  in 
Tables  12  and  14,  values  of  capacity  and  depth 
may  be  found  corresponding  to  selected  values 
of  mean  velocity.  Such  values  were  derived 
for  mean  velocities  of  2,  3,  and  4  ft./sec.,  and 
from  them  were  computed  the  42  values  of  hv 
in  Table  57. 

TABLE  57. —  Values  of  IAV,  the  synthetic  index  of  relative 
variation  for  capacity  in  relation  to  depth  of  current,  when 
mean  velocity  is  constant. 


Grade. 

vm 

Value  of  Uv  for  width  (feet)  of— 

0.23 

0.44 

0.66 

1.00 

1.32 

1.96 

(A) 
(B) 

(C) 
(D) 

(E) 
(G) 
(II) 

3 
4 

2 
3 
4 

2 
3 
4 

2 
3 
4 

2 
3 

3 
4 

3 

-0.32 

-0.75 
-  .69 

+  .32 
+  .19 

+0.09 
-  .14 
.34 

-0.26 
-1.02 

+  .25 
+  .18 
—    37 

-0.26 

-0.48 

-  .75 

0 
57 

+  .21 
36 

+  .22 
32 



+  .09 

-  .34 
-  .22 

-  .25 

-  .27 
-  .75 
-1.50 

-  .35 

+  .08 
-  .92 

-1.50 
—  .44 

-1.03 

—  .93 

-1.08 
-2.69 

75 

-1.35 
-1.19 

-s'i?" 

Most  of  these  values  are  negative,  but  nine 
are  positive.  The  mean  of  the  42  values,  com- 
bined with  regard  to  sign,  is  —0.54.  A  posi- 
tive value,  considered  by  itself,  indicates  that 
capacity  varies  directly  with  depth,  and  a  nega- 
tive value  that  the  variation  is  inverse.  At 


166 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


least  two  interpretations  may  be  considered — 
(1)  that  the  actual  variation  is  usually  inverse 
but  under  some  conditions  is  direct;  (2)  that 
the  law  of  inverse  variation  is  general  and  that 
the  apparent  exceptions  are  due  entirely  to  the 
imperfection  of  the  data.  The  second  inter- 
pretation accords  the  better  with  the  analogies 
afforded  by  other  branches  of  the  subject  and 
is  provisionally  accepted. 

The  interpretation  directs  attention  to  the 
general  discordance  of  the  computed  values  of 
Idr,  and  a  suggestion  may  be  made  as  to  the 
cause  of  that  discordance.  First,  the  measure- 
ment of  depth  was  the  most  difficult  of  the 
direct  measurements  performed  in  the  labora- 
tory, and  it  was  peculiarly  subject  to  possi- 
bilities of  systematic  error.  Second,  the  range 
of  slopes  through  which  depth  could  be  ob- 
served was  much  less  than  in  the  case  of 
capacity,  and  this  made  the  work  of  adjust- 
ment less  satisfactory.  Third,  in  the  compu- 
tation of  this  particular  index  the  depth  data 
enter  twice,  and  they  enter  in  such  way  that 
their  errors  cumulate  instead  of  canceling. 

In  Table  58  the  relations  of  I&  to  various 
conditions  are  shown  by  the  comparison  of 
corresponding  means.  1&  is  seen  to  vary 
directly  with  mean  velocity,  and  it  may  fairly 
be  inferred  to  vary  inversely  with  fineness,  but 
the  data  as  to  width  of  channel  are  contradic- 
tory and  inconclusive. 

The  control  of  the  index  by  mean  velocity 
affords  information  as  to  its  relation  to  slope 
and  discharge.  Mean  velocity  varies  directly 
with  both  slope  and  discharge,  and  it  does  not 
change  without  corresponding  change  in  at 
least  one  of  these  factors.  It  follows  that  the 
index,  which  varies  directly  with  mean  velocity, 
also  varies  directly  with  at  least  one  of  the 
factors  slope  and  discharge.  In  the  groups  of 
data  from  which  values  of  the  index  were  com- 
puted, increase  of  velocity  was  associated 
either  with  increase  of  slope  or  with  increase  of 
both  slope  and  discharge  but  in  no  case  with 
decrease  of  slope  or  discharge.  It  seems,  there- 
fore, proper  to  infer  that  I&  varies  in  magni- 
tude directly  with  both  slope  and  discharge. 
In  this  respect  it  stands  as  an  exception  among 
the  indexes  of  relative  variation  connected 
with  traction.  All  other  species  of  7  and  also  i 
vary  inversely  with  slope  and  discharge.  As  a 
check  on  this  exceptional  result,  certain  prob- 


able errors  have  been  computed.  In  the  upper 
division  of  Tablo- 58 ' -0.19,  the  mean  of  12 
values  of  7^  with  *a  velocity  of  2  ft./sec.,  is 
compared  with  —0.41,  the  corresponding  mean 
for  a  velocity  of  3  ft./sec.  The  difference 
between  these  means,  —0.22,  has  a  probable 
error  of  ±0.13.  A  similar  difference,  appear- 
ing in  the  comparison  of  indexes  for  velocities 
of  3  and  4  ft./sec.,  is  -0.24  ±0.18.  As  the 
two  differences  give  testimony  of  the  same 
tenor,  their  joint  evidence  is  stronger  than 
that  of  cither  separately,  so  that  the  discussion 
of  the  residuals  leaves  the  presumption  in  favor 
of  the  conclusion  that  this  particular  index 
constitutes  a  real  exception  in  its  relation  to 
slope  and  discharge. 

TABLE  58. — Partial  means  based  on  Table  57,  illustrating  the 
control  of  I^v,  by  mean  velocity,  fineness  and  width. 


Num- 
ber of 
values. 

Mean  of  Idv  for  mean  velocity  (ft./sec.)  of— 

2 

3 

4 

12 
9 

2 
10 

7 
2 
5 

1 

1 
2 
9 
8 
5 

-0.19 

-0.41 
-  .57 

-0.81 

Mean  of  far  for  grade— 

(A) 

(B) 

(C) 

(D) 

(E) 

(F)       (G)        (H) 

-0.54 

+0.18 
-   .11 

—0.32 

—  .15 

0  57 

51 

0  98 

—  .  :9 

..  -l.BO    .. 

1  08       0  75 

Mean  of  lav  far  width  (feet)  of— 

0.23 

0.44 

0.66 

1.00 

1.32             1.96 

-0.26 

-0.48 
61 

0  07 

—  .64 

—0.58 

-  .49 

0  61 

..    ..                  —0.44 

THE  THREE  CONDITIONS  COMPARED. 

When  depth  is  increased  without  change  of 
slope  (or  width  or  grade  of  d6bris),  its  increase 
is  effected  by  increase  of  discharge,  with  the 
result  that  capacity  is  increased,  so  that  ca- 
pacity is  an  increasing  function  of  depth. 
When  depth  is  increased  without  change  of 
discharge,  its  increase  is  effected  by  reducing 
slope,  with  the  result  that  capacity  is  reduced, 
so  that  capacity  is  a  decreasing  function  of 
depth.  When  depth  is  increased  without 
change  of  velocity,  its  increase  requires  increase 


RELATION  OF  CAPACITY  TO  DEPTH. 


167 


of  discharge  accompanied  by  diminution  of 
slope;  and  as  these  changes  have  opposite  in- 
fluences on  capacity,  it  is  not  evident  a  priori 
whether  capacity  will  be  enlarged  or  reduced. 
The  experimental  data  show  that  it  is  slightly 
reduced,  so  that  capacity  is  a  decreasing  func- 
tion of  depth. 

When  depth  is  reduced  without  change  of 
slope,  and  the  reduction  is  continued  progres- 
sively, a  stage  is  eventually  reached  in  which  the 
velocity  is  no  longer  competent  for  traction. 
It  is  probable,  therefore,  that,  under  this  con- 
dition, an  approximate  formula  for  C=f(d) 
might  involve  a  depth  constant  and  be  similar 
to  the  formula  (64)  used  for  0=f(Q). 

Reduction  of  depth  without  change  of  dis- 
charge involves  increase  of  velocity,  and  it  is 
evident  that  competence  does  not  lie  in  that 
direction.  But  increase  of  depth  involves  re- 
duction of  velocity  and  leads  eventually  to  a 
competent  velocity.  The  limiting  depth  cor- 
responding to  competence  is  therefore  a  great 
depth  instead  of  a  small  one.  As  mean  ve- 
locity now  varies  inversely  with  depth,  a  coor- 
dinate formula  might  take  the  form 


c=. 


B  being  used  as  a  general  constant  and  d  as  a 
constant  depth. 

When  depth  is  reduced  without  change  of 
mean  velocity,  the  efficiency  of  the  mean 
velocity  is  enhanced  and  competence  is  not 
approached.  When  depth  is  increased,  the 
efficiency  of  the  unchanged  mean  velocity  is 
diminished  and  a  (large)  competent  depth 
may,  under  some  conditions,  be  realized. 

To  show  the  numerical  relations  of  the  in- 
dexes by  which  these  three  capacity-depth  func- 
tions are  severally  characterized,  certain  means 
are  assembled  in  Table  59.  It  was  not  found 
possible  to  procure  values  of  the  different  in- 
dexes representing  closely  the  same  conditions, 
and  what  was  done  was  to  derive  for  each  index 
the  mean  of  all  determinations  made  for  each 
particular  grade  of  debris. 

The  arrangement  by  grades  points  again  to 
the  fact  that  all  the  indexes  vary  inversely  with 
fineness  of  debris,  but  the  rate  of  variation  is  in 
fact'  somewhat  greater  than  these  series  of 
values  suggest.  The  observations  on  the 
coarser  grades  were  made  with  steeper  average 


slopes  and  larger  average  discharges  than  those 
on  the  finer  grades,  and  the  effect  of  high  slopes 
and  large  discharges  (except  in  case  of  !&}  is 
to  reduce  the  indexes  of  relative  variation. 

TABLE  59. — Comparison  of  synthetic,  indexes  of  relative  vari- 
ation for  capacity  and  depth,  under  the  several  conditions  of 
constant  discharge,  constant  slope,  and  constant  mean 
velocity. 


Gride. 

Number  of  sepa- 
rate determina- 
tions. 

Means. 

Id<t 
Ids 

itn 

Ida 

IdV 

ftq 

Ids 

IdV 

(A) 
(B) 
(C) 
(D) 

Si 

(F) 
(0) 
(H) 

5 
18 
19 
9 
5 
5 
9 
3 

2 
17 
17 
10 
6 
4 
9 
3 

3 
12 
12 
7 
2 

1 

-  3.62 
-4.24 
-3.58 
-  4.39 
-  5.17 
-  8.73 
-  9.45 
-10.01 

2.29 
1.95 
2.41 
2.12 
1.58 
2.72 
2,81 
2.51 

-0.59 
-  .15 
-  .33 
—  .56 
-  .98 

-i'so 

-  .75 

-1.57 
-2.17 
-1.49 
-2.07 
-3.29 
-3.21 
-3.36 
-3.99 

73 

68 

42 

-5.28 

2.25 

-  .56 

-2.34 

From  the  general  means  at  the  bottom  of  the 
table  it  appears  that,  for  the  range  of  laboratory 
conditions,  capacity  is  2.34  times  as  sensitive 
to  the  control  of  depth  when  the  limiting  con- 
dition is  constant  discharge  as  when  the  limit- 
ing condition  is  constant  slope,  and  about  nine 
times  as  sensitive  as  when  the  limiting  condi- 
tion is  constant  mean  velocity. 

One  of  the  results  of  the  discussion  is  to 
emphasize  the  importance,  when  considering 
the  relation  of  tractional  work  to  depth,  of 
sharply  discriminating  the  conditions  under 
which  depth  is  regarded  as  a  variable. 

So  far  as  the  variations  of  the  capacity-depth 
indexes  admit  of  formulation  in  the  symbols 
used  for  other  indexes, 


=  ft 


P) 


-.(84) 
(84a) 
(84b) 


COMPARISON   OF  CONTROLS   BY  SLOPE,   DIS- 
CHARGE, MEAN  VELOCITY,  AND  DEPTH. 

In  Chapter  V  the  general  sensitiveness  of 
capacity  to  slope  is  compared  with  that  of 
capacity  to  discharge  by  means  of  coordinate 
values  of  the  exponent  i.  In  Chapter  VII  the 
sensitiveness  to  slope  is  compared  with  sensi- 
tiveness to  mean  velocity  (with  discharge  con- 
stant) by  means  of  coordinate  values  of  /. 
The  two  methods  of  comparison  are  so  far 


168  TRANSPORTATION    OF   DEBRIS  BY   RUNNING   WATER. 

related  that  their  results  should  be  fairly  har-  measure  last  mentioned  has  been  compared 

monious.     In  the  present  chapter  the  same  with  a  coordinate  measure  of  the  sensitiveness 

measure  of  sensitiveness  to  mean  velocity,  con-  of  capacity  to  discharge,   thus  completing  a 

verted  by  change  of  sign  to  a  measure  of  sensi-  circle  and  affording  opportunity  for  a  checking 

tiveness  to  depth   (with  discharge  constant),  of  estimates  in  respect   to  consistency.     The 

has    been   compared   with   a  similar  measure  ratios  resulting  from  the  four  comparisons  are 

for  depth  with  slope   constant.    Finally,  the  as  follows: 

index  of  control  by  slope          _ 
index  of  control  by  discharge 

index  of  control  by  velocity  (discharge  constant)  _ 
index  of  control  by  slope 

index  of  control  by  depth  (slope  constant)  __,»., 
index  of  control  by  discharge 

index  of  control  by  velocity  (discharge  constant)  _     ._, 
index  of  control  by  depth  (slope  constant) 

The  combination,  by  multiplication,  of  the  first  and  second  equations  gives 

index  of  control  by  velocity  (discharge  constant)  _     . 
index  of  control  by  discharge 

The  combination  of  the  third  and  fourth  gives  an  equation  with  identical  first  member,  but 
the  second  member  is  3.79.     The  two  results  differ  by  8  per  cent. 


CHAPTER  IX.— EXPERIMENTS  WITH  MIXED   GRADES. 


ADJUSTMENT  AND  NOTATION. 

The  tractional  load  of  a  natural  stream  in- 
cludes particles  with  great  range  in  size.  The 
grades  of  debris  used  in  the  laboratory  had  nar- 
rowly limited  ranges.  Although  the  reasons 
for  this  limitation  were  believed  to  be  adequate, 
the  possibility  was  recognized  that  the  laws 
discovered  by  the  use  of  artificial  grades  might 
not  apply  without  modification  to  natural 
grades;  arid  in  view  of  this  possibility  a  series 
of  experiments  were  arranged  to  bridge  over 
the  interval  between  the  artificial  and  the  natu- 
ral in  this  particular  respect.  The  same  appa- 
ratus and  the  same  general  methods  being  used, 
observations  were  made  first  on  mixtures  of  two 
grades,  then  on  mixtures  of  three  or  more,  and 
finally  on  a  natural  combination  of  sizes. 

When  work  on  mixtures  was  begun,  need  was 
soon  found  for  a  modification  of  the  experi- 
mental procedure.  During  the  automatic  proc- 
ess of  adjusting  the  slope  to  the  load  the  cur- 
rent acted  unequally  on  the  components  of  the 
mixture,  carrying  forward  an  undue  share  of 
the  finer  part  and  depositing  an  undue  share,  of 
the  coarser.  To  escape  the  difficulties  intro- 
duced by  this  partial  re-sorting,  a  run  was  in- 
terrupted after  it  had  gone  far  enough  to  indi- 
cate the  approximate  slope,  the  debris  was 
taken  from  arrester  and  experiment  trough  and 
was  throughly  remixed,  and  then  the  approxi- 
mate slope  was  artificially  constructed  in  the 
trough,  after  which  the  run  was  continued. 


The  observational  data  are  contained  in  divi- 
sions (J)  and  (K)  of  Table  4.  The  values  of 
capacity  in  relation  to  slope  were  adjusted  in 
the  same  manner  as  with  individual  grades, 
except  that  each  value  of  a  for  an  adjusting 
equation  was  derived  from  the  data  of  the  par- 
ticular observational  series,  without  influence 
from  related  series.  The  adjusted  capacities 
are  recorded  in  Table  60,  and  with  them  are 
fractional  capacities  computed  for  the  several 
grades  composing  the  mixtures.  Table  17  con- 
tains the  constants  of  the  adjusting  equations 
and  also  series  of  values  of  the  index  of  relative 
variation. 

In  each  set  of  experiments  with  mixture  of 
two  grades  the  proportions  of  the  components 
were  varied.  The  usual  series  of  proportions 
was  approximately  4:1,  2:1,  1:1,  1:2,  and 
1  : 4.  To  denote  these  mixtures,  a  notation 
has  been  adopted  similar  to  one  employed  in 
chemistry;  for  example,  a  mixture  of  grades 
(B)  and  (F)  in  the  proportion  4 : 1  is  desig- 
nated (B4F,). 

The  proportions  of  some  of  the  mixtures  are 
told  accurately  by  the  subscript  figures;  those 
of  others  only  approximately.  The  theoretic 
or  standard  mode  of  apportionment  was  by 
weight  of  dry  material,  but  when  this  was  not 
convenient  the  material  was  weighed  or  meas- 
ured in  moist  or  saturated  condition,  and  the 
actual  ratio  for  dry  weight  was  afterward 
learned  by  computation.  The  actual  propor- 
tions are  given  in  percentages  in  Table  4  (J) . 


TABLE  60. — Adjusted  values  of  capacity  in  relation  to  slope,  for  mixtures  of  two  or  more  grades  of  debris  and  for  an  unsorted 

natural  alluvium,  based  on  data  of  Table  4  (J)  and  (K). 


Grade    .  . 

(A,C,) 

(A,Ci) 

(A,Gi) 

(A,G,) 

Q    ..                   

0.363 

0.363 

0.363 

0.363 

Component  .        .          

(A) 

(C) 

Total. 

(A) 

(G) 

Total. 

(A)     !     (G) 

Total. 

(A) 

(G) 

Total 

S 

Value  of  C. 

0.5 

16 
23 
30 
37 
44 
51 

65 
80 
95 

16 
23 
30 
37 
44 
51 

66 
81 
96 

32 
46 
60 
74 
88 
102 

131 
161 
191 

13 
22 
32 
44 

57 
71 

99 
131 
165 

4 
7 
11 
15 
19 
23 

33 
44 

55 

17 
29 
43 
59 
76 
94 

132 
175 
220 

13 
21 
29 
39 
49 
60 

85 
113 

10 
15 
19 
24 
30 

42 
56 

20 
31 
44 
58 
73 
90 

127 
169 

8 
11 
14 
18 
22 

9 
12 
15 
19 
23 

17 
23 
29 
37 
45 

.6  

.7 

8 

.9  ...   .                                                   

1.0 

1.2 

32 
44 

57 

33 
45 

58 

65 
89 
115 

1.4  .. 

I  6 

1  8 

2.0...                                                   

2.8 

4  6 

i 

169 


170 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


TABLE  60.— Adjusted  values  of  capacity,  in  relation  to  slope,  for  mixtures  of  two  or  more  grades  of  debris  and  for  an  unsorted 
natural  alluvium,  based  on  data  of  Table  4  (J)  and  (K) — Continued. 


Grade 

(A,G,) 

(A!G() 

(B,Fi) 

(B,F,) 

Q  

0.363 

0.363 

0.363 

0.363 

(A) 

(G) 

Total. 

(A) 

(G) 

Total. 

(B) 

(F) 

Total. 

(B) 

(F) 

Total. 

s 

Value  of  r. 

0.5... 

26 
37 
48 
60 
72 
83 

107 
133 

7 
10 
14 
17 
20 
23 

30 
37 

33 

47 
62 
77 
92 
106 

137 
170 

22 
29 
38 
47 
54 
62 

78 
93 
109 

12 
17 
21 
25 
30 
35 

43 
52 
61 

34 
46 
59 
72 
84 
97 

121 
145 

170 

.6... 

| 

.7  

.8... 

.9...   . 

1.0 

7 

10 
14 
19 

14 

21 
28 
37 

21 

•    31 
42 
56 

3 

4 
5 
7 
9 
12 

9 

14 

20 
26 
34 
42 

12 

18 
25 
33 
43 
54 

1.2 

1.4                                                            ... 

1  6 

1.8..    . 

2  0 

Probable  error  (per  cent)  

2.1 

1.1 

Grade                                                        .  .  . 

(B.F,) 

(B,F,) 

(BiF4) 

(C«E,) 

Q                                                      

0.363 

0.363                                     0.363 

0.363 

(B) 

(F) 

Total. 

(B) 

(F) 

Total. 

(B) 

(F) 

Total. 

(H 

(E) 

Total. 

Value  of  C. 


05 

6 

29 

8 

37 

7                                                             

22 

24 

46 

2 

11 

13 

40 

11 

51 

.8                                

26 

30 

56 

9 

19 

28 

13 

16 

51 

14 

65 

.9  

31 

35 

66 

11 

23 

34 

4 

15 

19 

62 

17 

79 

10                                          

37 

41 

78 

12 

28 

40 

5 

18 

23 

74 

20 

94 

1  2                                            

48 

55 

103 

17 

37 

54 

6 

25 

31 

97 

26 

123 

14       

61 

68 

129 

21 

48 

69 

7 

32 

39 

122 

32 

154 

1.6  

75 

S3 

158 

26 

59 

85 

9 

40 

49 

146 

39 

185 

1  8 

89 

100 

189 

171 

46 

217 

0.1 

0.7 

Grade                

(CiEi) 

(CjEi) 

(CiEi) 

(CiEi) 

Q              

0.182 

0.363 

0.182 

0.363 

(C) 

(E) 

Total. 

(C) 

(E) 

Total. 

(C) 

(E) 

Total. 

(C) 

(E) 

Total. 

nt  r< 

0  5 

.6                       

20 

11 

31 

7 

ts 

15 

43 

14 

15 

29 

.8 

36 

19 

55 

19 

21 

40 

g 

44 

24 

68 

12 

12 

24 

25 

26 

51 

1  0 

53 

28 

81 

14 

15 

29 

30 

32 

62 

1.2 

70 

37 

107 

19 

20 

39 

42 

44 

86 

1  4 

88 

47 

135 

24 

26 

50 

53 

57 

110 

1.6 

44 

24 

68 

106 

57 

163 

29 

32 

61 

65 

71 

136 

1.8  .   . 

55 

29 

84 

124 

67 

191 

36 

38 

74 

78 

84 

162 

2.0 

66 

35 

101 

42 

46 

88 

90 

97 

187 

2.2 

77 

42 

119 

53 

58 

111 

0.7 



1.8 

2.3 

EXPERIMENTS   WITH    MIXED   GRADES. 


171 


TABLE  60. — -Adjusted  values  of  capacity,  in  relation  to  slope,  for  mixtures  of  two  or  more  grades  of  debris  and  for  an  unsorted 
natural  alluvium,  based  on  data  of  Table  4  ( J)  and  (AT)— Continued. 


Grade  

(C,E,) 

(CiE, 

(C!E.) 

(C,E() 

Q  

0.182 

0.363 

0.182 

0.363 

Component  

(C) 

(E) 

Total. 

(C) 

(E) 

Total. 

(C) 

(E) 

Total. 

(C) 

(E) 

Total. 

S 

Values  of  C. 

0.5... 

.6  

.7... 

8 
11 

13 
16 

22 
28 
34 
42 
48 
56 

17 
23 
29 
35 

48 
63 
77 
92 
108 
124 

25 
34 
42 

51 

70 
91 
111 
134 
156 
180 

.8  

.9  

3 

4 

5 
6 
8 
9 
11 
12 

13 

15 

20 
26 
32 
39 
45 
53 

18 

19 

25 
32 
40 
48 
56 
65 

6 

8 

11 
14 
18 
21 
25 

28 
33 

46 
61 
76 
92 
109 

34 

41 

57 
75 
94 
113 
134 

1.0.    . 

7 

9 
12 
15 
18 
22 
25 

15 

21 
27 
34 
41 
48 
55 

22 

30 
39 
49 
59 
70 
80 

1.2  

1.4  

1.6     .    . 

1  8 

2.0  

2.2  

Probable  error  (per  cent)  

2.2 

5  6 

0.1 

1.6 

Grade 

(C,G,) 

(C.G,) 

(C,G,) 

(C,G.) 

Q  

0.363 

0.363 

0.363 

0.363 

(C) 

(G) 

Total. 

(C) 

(G) 

Total. 

(C) 

(G) 

Total. 

(C) 

(G) 

Total. 

S 

Values  of  C. 

0.5... 

17 
24 
32 
42 
54 
65 

93 
126 
164 
208 

4 
6 
8 
10 
13 
16 

23 
32 
41 
52 

21 
30 
40 
52 
67 
81 

116 
158 
205 
260 

11 
19 
27 
36 
45 
55 

74 
96 
119 
141 

6 
9 
14 
18 
23 
27 

37 
48 
59 
71 

17 
28 
41 
54 
68 
82 

111 
144 
178 
212 

.6  

10 
15 
21 

27 

40 
53 

67 
80 

10 
16 
22 
28 

40 
53 
67 
«1 

20 
31 
43 
55 

80 
106 
134 
161 

.8  

.9  

4 
6 

11 
17 
23 
31 

8 
12 

22 

33 
47 
61 

12 

18 

33 
50 
70 
02 

1.0 

1  2 

1.  4 

1.6  

1.8... 

2.0  

Probable  error  (per  cent)  

1.9 

1.2 

4.4 

2.2 

Grade  

(E,Gi) 

(ESG,)                                  (E!G,) 

(EiG.) 

Q 

0.363 

0.363                                     0.363 

0.363 

(E) 

(G) 

Total. 

(E) 

(G) 

Total. 

(E) 

(G) 

Total. 

(E) 

(G) 

Total. 

S 

Value  of  C 

0.6 

10 
13 

17 
21 
25 

35 
47 
60 
75 

2 
3 
4 

5 
6 

9 
12 
15 
19 

12 
16 
21 
26 
31 

44 
59 
75 
94 

.7 

6 
9 
13 

18 

29 
43 
59 

3 

5 
7 
9 

15 
22 
30 

9 
14 
20 

27 

44 

65 
89 

8 

8 
10 
13 

18 
24 
31 

37 

8 
10 
13 

19 
25 
31 
38 

16 
20 
26 

37 
49 
62 
75 

.9 

4 

6 

8 
11 
14 
18 
23 

9 

11 

16 
22 
29 
37 
45 

13 

17 

24 
33 
43 
55 
68 

1.0  

1.2. 

1.4 

1.6     . 

1  8 

2.O..     . 



1.4 

172                                               TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 

TABLE  60.  —  Adjusted  values  of  capacity,  in  relation  to  slope,  for  mixtures  of  two  or  more  grades  of  debris  and  for  an  unsorted 
natural  alluvium,  based  on  data  of  Table  4  (  J)  and  (K)  —  Continued. 

Grade                                  !                (AiCiGj) 

(CuDssEuFeGj) 

Q                                                             0.  363 

0.182 

0.365 

Component  (A)      (C)      (G)     Total. 

(C) 

(D)      (E) 

(F)      (( 

5)      Total. 

(C) 

(D)          (E)          (F)          (G)        Total. 

S                                                                                                      Value  of  C. 

0.6... 

7                                                      2259 

8.2 
10.6 
12.8 
15.2 

20.3 
26.6 
32.8 
39.6 
47.3 

6.  6       2.  3 
8.  2       2.  8 
10.0       3.4 
11.8       4.1 

16.0      5.5 
20.  6       7.  1 
25.  6       8.  8 
30.  8     10.  6 
36.  8     12.  6 

1.1       0 
1.4 
1.7 
2.0 

2.7 
3.5       1 
4.4       1 
5.3       1 
6.3       2 

.3        18.9 
.5         23.5 
.6         28.  4 
.7         33.9 

.9        45.8 
.2        59 
.5         73 

.8         88 
.  1        105 

24.2 
29.2 
35.0 
40.9 

53.5 
67.5 
81.9 
95.4 

18.9          6.5          3.2           1.1              54 
22.8          7.8          3.9           1.3              65 
27.3           9.4           4.7           1.6              78 
31.8         10.9           5.4            1.8               91 

41.6         14.3           7.1          2.4             119 
52.5         18.0          9.0          3.0            150 
63.7         21.8          10.9           3.6             182 
74.2        25.4         12.7          4.2            212 

.8   .                             ...          4           4           9             17 

.9  7          7         13            27 

1.0                              ...             9          9        20            38 

1.2                                                16         16         33            65 

1.4  24         24         50             98 

1.6                                                    34         34         69           137 

1.8  45         45         91            181 

Probable  error  (per  cent)  4.  0 

0  4 



(CsDKE.jFsG,).                                                Natural. 

1 

Q 

0.545 

0.182 

0.363 

(C) 

(D) 

(E) 

(F) 

(0)           Total. 

S 

Values  of  C. 

0.4... 

17 
26 
36 

47 
60 
75 
91 

126 
168 
213 

.5  

.6.. 

36.4 
44.1 

50.4 
59.4 
68.0 

85.0 
103 
121.5 

28.4 
34.3 
39.9 
46.2 
52.8 

66.1 
80.2 
94.5 

9.7 
11.8 
13.7 
15.8 
18.1 

22.7 
27.5 
32.4 

4.8 
5.9 
6.8 
7.9 
9.0 

11.3 
13.7 
16.2 

1.6                81 
2.0                98 
2.  3              114 
2.  6               132 
3.0              151 

3.  8              189 
4.6              229 
5.4              278 

.7  

15 
20 
26 
33 

47 
63 
80 
98 
117 

138 
159 
181 

.8... 

.9  

1.0 

1  2 

1.4  

1.6 

1  8 

2.0  

2  2 

2.4  

2.6  -. 

Probable  error  (per  cent)  

0  4 

2.6 

0.8 

MIXTURES  OF  TWO  GRADES. 

The  relation  borne  by  the  traction  of  mix- 
tures to  the  traction  of  separate  grades  is  most 
clearly  shown  by  the  records  from  varied 
combinations  of  two  grades  only.  In  all 
experiments  with  mixtures  the  same  width  of 
channel,  1  foot,  was  used;  and  from  the  ad- 


justed capacities  for  mixtures  of  two  it  is 
possible  to  select  a  full  set  associated  with  the 
same  discharge,  0.363  ft.3/sec.,  and  the  same 
slope,  1.4  per  cent.  These  capacities  are 
arranged  for  comparative  examination  in 
Table  61,  and  to  them  are  added,  from  Table 
12,  the  corresponding  capacities  for  the  separ- 
ate grades. 


TABLE  61. — Capacities  for  traction,  with  varied  mixtures  of  two  grades. 


Grade. 

(A) 
(A3Gi) 

(A,Oij 

(A.Gj) 
(AiO,) 
(G) 

Capacities. 

Grade. 

Capacities. 

Grade. 

Capacities. 

Grade. 

Capacities. 

Grade. 

Capacities. 

Total. 

(A) 

(G) 

Total. 

(C) 

(G) 

Total. 

(B) 

(F) 

Total. 

(C) 

(E) 

Total. 

(E) 

(G) 

185 
175 
169 
89 
42 
25 
16 

185 
131 
113 
44 
14 
5 
0 

0 
44 
56 
45 
28 
20 
16 

(C) 
(C'lo!) 

(0,0,5 

(CiGs) 
(G) 

143 
158 
144 
106 
50 
16 

143 
126 
96 
63 
17 
0 

0 
32 
48 
63 
33 
16 

IBJ 

(B.Ft) 
<B?F|) 

(BiFj) 

cb 

149 
170 
145 
129 
69 
39 
32 

149 
133 
93 

61 
21 
7 
0 

0 
37 
52 
68 
48 
32 
33 

iF$ 

(C^tii 

(CiE, 
(C,E, 
(C,E, 

(^E)° 

143 
154 

135 
110 

91 
75 
62 

143 

122 
88 
53 
28 
14 
0 

0 
32 

47 
57 
63 
61 
.  62  ' 

CBiOi] 

(E,G,) 

62 
59 
66 
49 
33 
16 

62 

47 
43 
24 

11 
0 

0 
12 
22 
25 
22 
16 

EXPERIMENTS    WITH    MIXED   GRADES. 


173 


Each  vertical  column  of  capacities  shows  the 
tractional  power  of  the  current,  first  for  the 
finer  component  alone,  then  for  mixtures  with 
progressively  increasing  shares  of  the  coarser 
component,  and  finally  for  the  coarser  alone. 
The  order  of  the  different  groups  is  that  of  the 
contrast  in  fineness  between  the  finer  and 
coarser  components.  If  the  linear  fineness  of 
the  finer  be  divided,  in  each  case,  by  the  fine- 
ness of  the  coarser,  the  ratios  obtained  are 
(AG)  (CG)  (BF)  (CE)  (EG) 
16.2  9.7  8.5  3.4  2.9 


CA)' 


(G) 


(E) 


(G) 


0 
100 


50 
50 


100 
0 


FIGURE  55.— Tractional  capacity  for  mixed  debris,  in  relation  to  propor- 
tions of  component  grades. 

The  data  of  the  table  are  plotted,  with  a 
different  arrangement,  in  figures  55  and  56. 
In  figure  55  the  total  capacities  are  plotted  in 
relation  to  the  proportions  of  fine  and  coarse 
in  the  mixtures.  The  vertical  scale  being  of 
capacity,  the  horizontal,  if  read  from  right  to 
left,  is  of  percentage  of  the  finer  component, 
or  if  read  from  left  to  right,  is  of  percentage  of 
the  coarser.  Figure  56  (p.  1 74)  shows,  in  parallel 
columns,  the  capacities  for  the  component 
grades,  those  for  the  coarser  components  being 
at  the  left. 

Attention  may  first  be  directed  to  the  total 
capacity  curves  of  figure  55.  If  the  capacities 


for  mixtures  were  related  in  the  simplest 
manner  to  the  proportions  of  components,  the 
curve  of  capacities  would  be  a  straight  line 
joining  the  points  (at  the  extreme  left  and 
extreme  right)  given  by  the  capacities  for  the 
components.  The  (EG)  curve — that  placed 
lowest  in  the  diagram — lying  wholly  above  such 
a  hypothetic  line,  would  suggest  that  all  mix- 
tures give  an  advantage  in  traction,  but  this 
suggestion  is  not  supported  by  the  other 
curves.  The  upper  curve,  for  example,  would 
lie  as  much  below  as  above  a  straight  line 
joining  its  extremities.  The  capacity  for  the 
mixture  appears  always  to  exceed  what  may 
be  called  the  pro  rata  estimate  when  the  finer 
component  has  the  higher  proportion,  but  it 
may  fall  below  that  estimate  when  the  coarser 
component  predominates. 

The  curves  of  figure  56  show  the  capacities 
apportioned  to  the  components  of  the  mix- 
tures. Each  one  represents  that  portion  of 
the  total  load  which  consists  of  the  material 
of  one  component.  In  each  curve  of  the  left- 
hand  column  the  ordinate  at  the  right  repre- 
sents the  load  of  the  coarser  material  when  by 
itself,  and  the  successive  ordinates  toward  the 
left  show  how  the  load  is  modified  by  the 
admixture,  in  the  material  fed  to  the  current, 
of  gradually  increasing  percentages  of  the 
finer  debris.  There  are  no  observations  with 
very  small  percentages  of  the  finer  material, 
but  in  each  case  the  second  fixed  point  of  the 
curve  shows  an  increase  of  load.  The  addition 
of  the  finer  d6bris  not  only  increases  the  total 
capacity  but  increases  the  capacity  for  the 
coarser  d6bris.  The  amount  of  the  latter 
increase  appears  to  be  greater  as  the  contrast 
in  fineness  of  components  is  greater,  and  in 
the  extreme  case  the  capacity  for  the  coarser 
is  multiplied  by  3.5.  The  capacity  of  the 
current  for  debris  of  grade  (G)  is  16  gm./sec.; 
but  when  that  debris  is  mixed  with  twice  its 
weight  of  grade  (A),  which  is  16.2  times  as 
fine,  the  capacity  for  the  mixture  is  so  large 
that  one-third  of  it,  apportioned  to  grade  (G), 
is  56  gm./sec.  In  three  of  the  five  suites  of 
experiments  the  highest  capacity  recorded  for 
the  coarser  debris  corresponds  to  the  mixture 
of  1 : 1 .  In  the  others  it  corresponds  approxi- 
mately to  2:1  and  1:2.  If  the  position  of  the 
maximum  is  related  to  contrast  in  fineness,  it 
is  associated  with  a  larger  ratio  of  the  finer 
component  when  the  contrast  is  great. 


174 


TRANSPORTATION    OP   DEBRIS   BY   RUNNING   WATER. 


The  corresponding  curves  for  the  finer  com- 
ponent (at  the  right  in  fig.  56)  show  that 
capacity  for  finer  de'bris  is  influenced  by  the 
admixture  of  coarser,  but  not  in  the  same 
way.  Each  addition  of  the  coarser  reduces 
capacity,  the  rate  of  reduction  being  at  first 
gradual,  then  more  rapid,  and  afterward 
gradual. 

The  general  effect  of  the  addition  of  fine 
material  being  to  increase  capacity,  and  the 
effect  of  adding  coarse  material  to  reduce 
capacity,  let  us  now  inquire  the  effect  of 


diversifying  the  material  of  the  load  by  ex- 
changing part  of  it  for  a  finer  grade  and  part 
for  a  coarser.  Probably  no  general  answer  to 
this  question  may  be  derived  from  our  data, 
but  a  partial  answer  is  possible  if  we  assume 
that  the  initial  grade  is  separated  from  each 
of  the  substituted  components  by  the  same 
contrast  of  fineness,  expressed  as  a  ratio. 
To  take  a  concrete  example,  from  Table  61, 
the  capacity  for  grade  (A)  is  185  gm./sec., 
for  grade  (G)  16  gm./sec.,  and  for  (A,Gj), 
their  equal  mixture,  89  gm./sec.  When  for 


(£> 


..(G) 


IE)- 


FIGURE  K.— Tractional  capacities  of  components  of  mixed  grades,  in  relation  to  the  percentages  of  the  components  in  the  mixtures.    Curves 
at  left  show  capacities  for  coarser  component,  at  right  for  finer.    Katios  at  left  show  percentages  of  finer  component ,  at  right  of  coarser. 


half    of    (A)    there   is   substituted    an    equal 
amount  of  (G),  which  is  16.2  times  coarser, 

89 
the  capacity  is  changed  in  the  ratio 


When  for  half  of  (G)  there  is  substituted  an 
equal  amount  of  (A),  which  is  16.2  times  finer, 

89 
the  capacity  is  changed  in  the  ratio  .-^  =  5.56. 

The  geometric  mean  of  these  ratios,  1.64,  may 
plausibly  stand  for  the  effect  of  substituting 
an  equal  mixture  of  (A)  and  (G)  for  a  grade 
symmetrically  intermediate  between  (A)  and 
(G).  The  method  is  easily  criticized,  and  its 
assumptions  will  certainly  not  bear  close 


scrutiny,  but  it  nevertheless  yields  a  sort  of 
composite  which  is  of  use  in  showing  that  the 
general  effect  of  diversifying  a  stream's  trac- 
tional  load  is  to  enlarge  capacity.  Corre- 
sponding composite  ratios  have  been  obtained 
from  the  other  examples  of  Table  61  and  are 
given  below. 


Ratio  of 

Grade. 

Ratio  of 
fineness. 

capacity 
change 
attributed 

to  mixture. 

(AG 

10.2 

1.64 

(CG 

9.7 

2.22 

(BF 

8.5 

1.72 

(CE 

3.4 

1.17 

(EG) 

2.9 

1.56 

EXPERIMENTS  WITH  MIXED  GRADES. 


175 


The  composite  ratios  have  been  arranged  in 
the  order  of  the  ratio  of  fineness,  but  the  com- 
parison shows  no  correspondence.  They  prove 
equally  inharmonious  when  compared  with  the 
fineness  of  the  .finer  component,  the  fineness  of 
the  coarser  component,  or  the  fineness  of  the 
mixture.  Their  irregularities  must  be  as- 
cribed to  observational  errors  and  to  causes  not 
at  present  to  be  discriminated  from  observa- 
tional errors.  It  is  of  interest,  however,  to  note 
that  the  large  measure  of  capacity  change  as- 
sociated with  the  (CG)  combination  might  be 
inferred  also  from  a  comparison  which  involves 
a  different  viewpoint  and  also  some  practically 
independent  data.  If  we  think  of  grades  (A), 
(C),  and  (E)  as  modifiers  of  capacity  for  grade 
(G),  we  may  compare  their  efficiencies  by  means 
of  the  following  quantities,  taken  from  Table  61 : 


Grade. 

Capacity. 

Grade. 

Capacity. 

(AiOO 

(C,G,) 
(EiG,) 

89 
106 
49 

(AiG,) 
(C.Gj) 
(EiGs) 

42 
50 
33 

The  superiority  of  grade  (C)  as  a  modifier  for 
(G)  is  thus  brought  out  without  making  use  of 
the  capacities  for  the  uncombined  grades  (A), 
(C),  (E),  and  (G);  and  the  result  from  mix- 
tures of  1 : 1  is  supported  by  that  from  mixtures 
of  1 : 2. 

CONTROL  BY  SLOPE  AND  DISCHARGE. 

The  preceding  comparisons  are  conditioned 
by  a  discharge  of  0.363  ft.3/  sec.,  a  slope  of  1.4 
per  cent,  and  a  channel  width  of  1  foot.  With 
a  different  set  of  conditions  a  different  set  of 
quantitative  relations  would  be  found,  and  the 
qualitative  also  would  doubtless  be  modified. 
The  observations  on  mixtures  included  no  other 
width,  and  there  was  but  a  single  set  of  ex- 
periments using  a  different  discharge,  but  the 
range  in  slope  was  coordinate  with  that  for  the 
separate  grades. 

Figure  57  shows  the  capacity-slope  curves 
for  the  (AG)  set  of  experiments,  figure  58  for 
the  (BF)  set,  and  figure  59  for  the  (CE)  set. 
In  figure  57  the  curves  for  mixtures  form  a 
graded  series  between  those  for  the  component 
grades,  and  there  is  almost  perfect  harmony  of 
form  and  attitude.  As  the  points  representing 
capacities  associated  with  a  slope  of  1.4  per 
cent  all  lie  in  the  same  vertical  line,  and  as 
similar  points  for  another  slope  lie  in  some 


other  vertical  line,  it  is  evident  by  inspection 
that  inferences  from  data  of  any  other  available 
slope  would  be  practically  identical  with  those 
from  the  slope  of  1 .4  per  cent.  It  is  also  evident 


200 


o 

ID 

Q- 


1:1 


0  I  2 

Slope 

FIGURE  57.— Curves  of  capacity  in  relation  to  slope  for  grade  (A), grade 
(G),  and  mixtures  of  those  grades.  The  ratios  of  components  in  the 
mixtures  are  indicated. 

that  indexes  of  relative  variation,  \  or  /„  for 
the  mixtures  constitute,  with  those  for  the 
components,  an  orderly  system.  The  same 
remarks  apply  also  to  the  (CE)  groups  of 


200 


100 


ifs> 


1:1 


1:2 


0  I  2 

Slope 

FIGURE  58.— Curves  of  capacity  in  relation  to  slope  for  grade  (B),  grade 
(F),  and  mixtures  of  those  grades.  The  ratios  of  the  components  in 
the  mixtures  are  indicated. 

curves,  but  they  do  not  apply  to  the  (BF) 
group  in  figure  58.  The  attitudes  of  the  curves 
for  mixtures  are  there  out  of  harmony  with  the 
attitudes  of  the  (B)  and  (F)  curves.  The 
curves  for  the  mixtures  seem  to  belong  to  a 


176 


TBANSPORTATION   OF   DEBBIS   BY   RUNNING   WATER. 


different  system,  intersecting  or  tending  to 
intersect  the  curves  for  the  components.  Evi- 
dently the  indexes  of  relative  variation  are 
inharmonious,  and  evidently  the  inferences 
drawn  from  data  for  the  slope  of  1.4  per  cent 
would  not  be  duplicated  by  a  discussion  of 
data  from  a  slope  of  1.0  per  cent  or  1.6  per  cent. 


200 


o 

(0 
Q. 
<U 

O 


100 


Slope 

FIGURE  59.— Curves  of  capacity  in  relation  to  slope  for  grade  (C),  grade 
(E),  and  mixtures  of  those  grades.  The  ratios  of  the  components  in 
the  mixtures  are  indicated. 

Graphic  comparison  has  been  extended  to  the 
remaining  data  of  Table  60,  so  far  as  dual  mix- 
tures are  concerned,  but  no  marked  discordance 
has  been  discovered  outside  of  the  (BF)  group. 
Although  the  cause  of  the  exceptional  discord- 
ance has  not  been  found,  I  believe  that  it 
should  be  ascribed  to  some  exceptional  though 
unknown  circumstance  and  not  be  permitted 
to  nullify  the  otherwise  harmonious  testimony. 
The  tenor  of  that  testimony  is  that  the  relation 
of  capacity  to  slope  is  substantially  the  same 
for  mixtures  as  for  simpler  grades  of  debris. 
There  is,  however,  a  noteworthy  qualification 
to  this  statement,  in  that  the  values  of  a  have 
a  smaller  average  for  mixtures  than  for  com- 
ponent grades. 

The  single  comparison  possible  between 
results  obtained  with  different  discharges  indi- 
cates that  with  mixtures,  just  as  with  their 
components,  capacity  increases  with  discharge 
in  more  than  simple  ratio. 

MIXTURES  OF  MORE  THAN  TWO  GRADES. 

Experiments  were  made  with  a  mixture  of 
three  grades,  and  with  a  mixture  of  five.  The 


former  was  observed  with  discharges  of  0.182 
and  0.363  ft.3/sec.,  the  latter  with  discharges  of 
0.182,  0.363,  and  0.545  ft.3/sec.  (See  Tables  4 
(J)  and  60.) 

Brief  consideration  only  will  be  given  to  the 
data  from  the  mixture  of  three,  (AjCA), 
because  the  points  on  which  they  bear  are  more 
fully  covered  by  the  data  from  the  mixture  of 
five.  Grades  (A)  and  (C)  differ  from  one 
another  in  fineness  much  less  than  either 
differs  from  grade  (G) .  The  triple  mixture  may 
therefore  be  thought  of  as  half  coarse  and  half 
fine,  with  the  distinction  that  the  fine  half  is 
made  up  of  two  grades.  By  comparing  data 
from  it  with  data  from  the  closely  related  mix- 
tures (AA)  and  (CA), m  which  the  fine  portion 
is  of  a  single  grade,  we  may  throw  light  on  the 
question  whether  the  advantage  to  traction 
which  is  obtained  by  substituting  two  grades 
for  one  may  be  augmented  by  further  diversi- 
fication. With  discharge  0.363  ft.3/sec.  and 
slope  1.4  per  cent  the  capacities  for  the  three 
mixtures  are 

(AA)         (AAG2)         (CA) 
89  98  106 

As  the  capacity  for  the  triple  mixture  has  a 
value  midway  between  those  for  the  two  dual 
mixtures,  no  advantage  is  indicated  for  the 
greater  diversification. 


Slope 


Slope 


FIGURE  60.— Curves  of  capacity  in  relation  to  slope  for  a  mixture  of 
five  grades,  (CDEFO).  Comparison  of  mixture  curves  for  three 
discharges,  and  of  mixture  curve  with  curves  for  component  grades. 

In  the  mixture  of  five  grades  the  proportions 
were  so  arranged  as  to  approximate  a  natural 
combination.  The  components  and  their  per- 
centages are  shown  by  writing  (C^D^E^FA)- 
The  curves  of  capacity  in  relation  to  slope  are 
shown,  for  the  three  discharges,  in  the  left- 


EXPERIMENTS    WITH    MIXED   GRADES. 


177 


hand  diagram  of  figure  60.  In  the  right-hand 
diagram  the  curve  for  discharge  0.363  ft.3/sec. 
is  repeated,  and  with  it  are  placed  the  corre- 
sponding curves  for  four  of  the  component 
grades — that  for  grade  (F)  being  omitted 
because  it  is  nearly  coincident  with  the  one  for 
grade  (E). 

The  curve  for  the  mixture  is  not  only  of  the 
same  type  with  the  others,  but  runs  nearly 
parallel  with  those  nearest.  For  all  slopes  the 
capacities  obtained  for  the  mixture  are  greater 
than  for  any  component.  The  fact  that  the 
mixture  gives  a  greater  capacity  than  does  the 
fine  grade  (C)  alone  shows  that  the  addition  to 
(C)  of  35  per  cent  of  (D)  and  20  per  cent  of 


still  coarser  grades  works  an  advantage  instead 
of  a  detriment. 

In  figure  58  it  is  seen  that  the  addition  of  20 
per  cent  of  coarser  d6bris  to  grade  (B)  in- 
creases capacity,  and  in  figure  59  that  a 
similar  increase  accompanies  the  addition  of 
20  per  cent  of  coarser  debris  to  grade  (C). 
So  far  as  the  case  of  the  more  complex  mixture 
is  comparable  to  these,  there  is  no  indication 
that  a  mixture  of  great  complexity  has  ad- 
vantage for  traction  over  a  mixture  of  two 
components  only.  As  to  this  point  the  in- 
ference from  data  of  the  five-part  mixture  is 
supported  by  that  already  drawn  from  a 
datum  of  the  (A,CtG2)  mixture,  and  it  is 


300- 


200- 


200 


O 

ra 

Q- 

co 

u 


100 


Slope 


fG>     - 


.4-  .8  1.2 

Discharge 


FIGURE  01.— Capacity-slope  curves  for  related  mixtures  and  capacity-discharge  curves  for  mixture  and  component  grades. 


further  supported  by  the  facts  brought  to- 
gether in  the  left-hand  diagram  of  figure  61, 
which  shows  the  capacity-slope  curves  of  mix- 
tures (C4Et)  and  (C4GJ  along  with  that  for  the 
five-part  mixture. 

The  right-hand  diagram  of  figure  61  is  a 
group  of  plots  of  capacity  as  a  function  of  dis- 
charge. These  plots  pertain  to  the  complex 
mixture  and  its  five  components,  and  all  are 
conditioned  by  a  slope  of  1.2  per  cent.  They 
show  that  the  fractional  superiority  of  the 
mixture  is  not  confined  to  the  use  of  a  particu- 
lar discharge,  and  they  indicate  also  that  the 
capacity-discharge  relation  is  essentially  the 
same  for  the  mixture  as  for  separate  grades. 
The  locus  of  C  =f(Q)  for  the  mixture  is  approxi- 
mately a  straight  line,  and  if  produced  it  inter- 
sects the  axis  of  Q  to  the  right  of  the  origin. 
It  might  be  expressed  by  an  equation  in  the 
form  of  (64)  with  an  exponent  near  unity. 

20021°— No.  80—14 12 


A  NATURAL  GRADE. 

Two  series  of  experiments  were  made  with  an 
alluvium  in  its  natural  condition,  except  that 
the  very  finest  constituents  had  been  removed 
by  passing  it  over  a  60-mesh  sieve.  The  obser- 
vations are  recorded  in  Table  4  (K),  and  the 
adjusted  capacities  in  Table  60.  In  figure  62 
the  capacity-slope  curves  are  plotted,  and  each 
is  accompanied,  for  comparison,  by  the  corre- 
sponding curves  for  grades  (A)  and  (C). 

The  approximate  mechanical  analysis  of  this 
material,  stated  in  terms  of  the  separated 
grades  of  tho  laboratory  series,  is  as  follows: 


Per  cent. 


(A) 
(B) 
(C) 


13 

27 


Per  cent. 

(D) 42 

(E) 10 

Coarser  than  (E) 2 

The  fact  that  the  capacities  (fig.  62)  are 
greater  than  those  for  grade  (C),  notwithstand- 
ing the  dominance  of  a  component  correspond- 


178 


TRANSPORTATION    OF   DEBRIS  BY   RUNNING   WATER. 


ing  to  the  coarser  grade  (D),  testifies  again  to 
the  advantage  for  traction  of  a  mixture  as 
compared  to  a  grade  of  narrow  range  in  fineness. 
It  will  be  observed  also  that  each  of  the 
curves  from  the  natural  grade  resembles  closely 
its  neighbors  from  artificial  grades.  So  far  as 
their  evidence  goes  the  type  is  the  same  for 
both,  and  the  tendency  of  their  evidence  is  to 
show  that  the  laws  connecting  capacity  with 
slope,  as  developed  by  the  study  of  sorted 
debris,  apply  also  to  unsorted  stream  alluvium. 


300- 


200 


100- 


fA) 


0  I  2 

Slope 

FIGURE  62.— Capacity-slope  curves  for  a  natural  grade  of  de'bris,  com- 
pared with  curves  for  sieve-separated  grades. 

Table  63  (p.  180)  gives  computed  finenesses 
for  various  mixtures  and  for  this  natural  grade 
of  de'bris.  The  fineness  of  the  natural  grade  is 
nearly  identical  with  that  of  grade  (C),  and  the 
two  thus  afford  a  direct  comparison  between  the 
capacities  of  a  natural  grade  and  a  narrowly 
limited  grade.  For  the  same  discharge  and 
slope  their  tabulated  capacities  are  respectively 
168  and  143  gm./sec.,  the  ratio  of  advantage  to 
the  natural  grade  being  1.17.  The  computed 
fineness  of  grade  (CDEFG)  does  not  corre- 
spond to  that  of  any  simple  grade,  but  a  com- 
parison made  by  means  of  interpolation  gives 
150  and  131  gm./sec.  as  corresponding  capaci- 
ties for  the  mixture  and  a  simple  grade  of  the 
same  mean  fineness,  and  the  ratio  of  the  first 
to  the  second  is  1.15.  These  ratios  are  smaller 
than  those  estimated  from  data  for  binary 
mixtures  (p.  174),  but  are  coordinate  in  value. 
The  question  of  relative  authority  will  be  con- 
sidered later. 

CAUSES  OF  SUPERIOR  MOBILITY  OF 
MIXTURES. 

When  a  finer  grade  of  debris  is  added  to  a 
coarser    the   finer   grains    occupy    interspaces 


among  the  coarser  and  thereby  make  the  sur- 
face of  the  stream  bed  smoother.  This  quality 
of  smoothness  appealed  to  the  eye  during  the 
progress  of  the  experiments.  One  of  the 
coarser  grains,  resting  on  a  surface  composed 
of  its  fellows,  may  sink  so  far  into  a  hollow  as 
not  to  be  easily  dislodged  by  the  current,  but 
when  such  hollows  are  partly  filled  by  the 
smaller  grains  its  position  is  higher  and  it  can 
withstand  less  force  of  current.  In  other 
words,  the  larger  particles  are  moved  more 
readily  on  the  smoother  bed,  and  this  fact  also 
was  a  matter  of  direct  visual  observation.  The 
promotion  of  mobility  applies  not  only  to  the 
starting  of  the  grain  but  to  its  continuance  in 
motion.  It  encounters  less  resistance  as  it 
rolls  or  skips  along  the  bed,  and  it  is  less  apt  to 
be  arrested.  When  a  single  large  particle 
travels  along  a  bed  composed  wholly  of  grains 
much  smaller  it  rarely  leaps,  but  rolls  instead, 
and  it  must  in  general  be  true  that  the  larger 
particles  in  mixtures  roll  more  and  skip  less 
than  their  smaller  companions. 

The  admixture  of  finer  debris  thus  changes 
the  mode  of  traction  for  the  coarser,  and  it  is 
believed  that  the  enhanced  capacity  is  due 
mainly  to  this  change.  Capacity  for  the 
coarser  is  increased  because  the  new  condition 
reduces  its  resistance  to  the  force  of  the 
current. 

The  fact  that  under  some  conditions  the 
capacity  for  fine  material  is  slightly  increased 
by  the  addition  of  coarser  is  not  so  easily  ex- 
plained. The  coarser  grams  do  not  make  the 
bed  smoother  but  rougher.  The  rougher  bed 
retards  the  current.  Even  while  rolling  the 
larger  grains  are  holding  back  the  water,  and 
the  larger  grains  reduce  the  area  of  bed  on 
which  the  traction  of  the  smaller  takes  place. 
In  these  ways  the  presence  of  the  coarse  mate- 
rial tends  to  reduce  the  capacity  of  the  current 
for  the  fine,  and  these  factors  certainly  seem 
adequate  to  explain  the  general  fact  that 
capacity  for  the  finer  debris  is  reduced  by  ad- 
mixture of  the  coarser. 

Two  factors  may  be  named  with  the  opposite 
tendency.  The  first  is  the  impact  of  the 
coarser  particles.  In  rolling  and  leaping  they 
disturb  the  finer,  tending  thus  to  dislodge  them 
from  their  resting  places  and  either  start  them 
forward  or  else  give  them  new  positions  from 
which  they  may  be  more  easily  swept.  The 
second  is  the  production  of  diversity  in  the 


EXPERIMENTS    WITH    MIXED    GRADES. 


179 


current.  Every  obstruction  diversifies  the 
current.  The  deflection  necessary  to  pass  it 
both  constitutes  and  causes  diversity  of  direc- 
tion, and  diversity  of  direction  necessitates 
diversity  of  velocity.  If  a  pebble  is  placed  on 
the  sandy  bed  of  a  small  stream,  the  trans- 
formation of  the  adjacent  parts  of  the  bed  by 
the  diversified  current  is  obvious.  The  build- 
ing up  of  the  bed  in  the  lee  of  the  pebble  testi- 
fies to  lowered  velocity,  and  the  scouring  at  the 
side  to  heightened  velocity.  In  the  same  way 
each  coarser  gram  of  a  heterogeneous  stream 
load  diversifies  the  current  about  it  and  gives 
to  such  of  the  filaments  as  are  accelerated 
greater  power  for  the  traction  of  finer  grains. 
With  reference  to  the  transportation  of  the 
finer  debris,  the  coarser  grains  have  the  func- 
tion of  obstructions  whether  they  are  partly 
embedded  or  lie  on  the  surface  or  are  rolled 
along. 

If  the  factors  concerned  in  the  traction  of  the 
finer  components  of  mixtures  have  been  cor- 
rectly stated,  it  is  not  difficult  to  understand 
that  under  most  conditions  the  net  result  of 
their  influences  will  be  a  reduction  of  capacity, 
and  also  that  special  conditions  may  deter- 
mine an  increase. 

VOIDS. 

The  packing  together  of  larger  and  smaller 
grains  which  tends  toward  a  smooth  stream 
bed  tends  also  toward  the  reduction  of  inter- 
stitial spaces  within  the  bedded  d6bris,  thus 
reducing  the  percentage  of  voids.  It  was  sug- 
gested in  the  laboratory  that  the  percentage  of 
voids,  used  inversely,  might  serve  as  a  sort 
of  index  of  mobility,  and  estimates  of  the  voids 
were  accordingly  made  for  most  of  the  mate- 
rials employed  in  the  experiments  with  mixed 
grades.  Partly  for  this  purpose,  and  partly  to 
obtain  the  factors  needed  to  correct  the  weigh- 
ings of  load  for  interstitial  water,  a  series  of 
special  weighings  were  made. 

A  vessel  holding  535  cubic  centimeters  was 
filled  with  saturated  debris  and  weighed.  Af- 
terward the  same  debris  was  weighed  in  a  dry 
condition.  The  computation  was  made  by  the 

formula 

W—  W 
Percentage  of  voids  =  — -„. 

in  which  W  is  the  weight  of  saturated  d6bris 
in  grams  and  W  the  weight  of  dry  d6bris. 

Table  62  contains  the  estimated  voids  for  the 
binary  mixtures.  In  each  series  the  percent- 


ages are  smaller  for  the  mixtures  than  for  the 
component  grades;  and  when  the  percentages 
were  plotted  in  relation  to  the  proportions  of 
component  grades  (after  the  manner  of  the 
capacities  in  figs.  55  and  56)  each  series  was 
found  to  indicate  a  minimum.  The  positions 
of  the  minima  correspond  to  mixtures  with  30 
to  40  per  cent  of  the  finer  grade  of  debris.  In 
comparing  voids  with  capacities,  the  minima 
of  the  void  curves  are  to  be  considered  in  rela- 
tion to  the  maxima  of  the  capacity  curves. 


Capacity 


Percentage 
of  voids 


Linear 
fineness 


Bulk 
fineness 


FIGURE  63.— Curves  showing  the  relations  of  various  quantities  to  the 
proportions  of  fine  and  coarse  components  in  a  mixture  of  two  grades 
of  ddbris,  (C)  and  (G).  The  horizontal  scale,  when  read  from  left  to 
right,  shows  the  percentage  of  the  coarser  component  in  the  mixture. 

Those  maxima,  however,  are  associated  with 
mixtures  having  60  to  90  per  cent  of  the  finer 
grade;  and  the  attempt  at  correlation  therefore 
fails.  A  single  void  curve  is  reproduced  in 
figure  63,  together  with  the  corresponding 
capacity  curve. 

TABLE  62. — Percentages  of  voids  in  certain  mixed  grades  of 
debris,  compared  with  the  percentages  in  the  component 
grades. 


Percent- 
age of 
finer 
grade  in 
mixture. 

Percentage  of  voids  in  grade  — 

(AC) 

(AG) 

<BF) 

(CE) 

(CO) 

(EG) 

0 
20 
83 
50 
67 
SO 
100 

44 

38 
28 
24 
29 
31 
37 
44 

37 
31 
29 
29 
34 
80 

46 

40 
33 
32 
33 
35 
37 
44 

38 

38 

26 
30 
33 

38 
44 

29 
28 
34 
36 

40 

40 

44 

180 


TRANSPORTATION   OF   DEBKIS   BY   RUNNING   WATER. 


FINENESS. 

A  similar  attempt  was  made  to  correlate  the 
capacity  curves  of  figures  55  and  56  with  fine- 
ness. In  computing  the  fineness  of  a  mixture, 
the  finenesses  of  its  components  were  used  as 
data,  and  the  combination  was  made  with  bulk 
finenesses  (p.  21)  as  follows:  Denoting  the  pro- 
portions of  components  by  a,  b,  c,  etc.,  their 
bulk  finenesses  by  F2',  F2",  F2",  etc.,  and  the 
fineness  of  the  mixture  by  F2m, 


•....(85) 


a  +  b  +  c  +  etc. 


Linear   fineness,    F,    was   then   computed   by 
formula  (88). 

The  results  are  listed  in  Table  63,  together 
with  the  corresponding  capacities  for  traction 
when  the  discharge  is  0.363  ft.3/sec.  and  the 
slope  is  1.4  per  cent. 

TABLE  63. — Finenesses  of  mixed  grades  and  their  components. 
[Computed  from  data  in  Tables  1, 4  (J),  and  4  (K).] 


Grade. 

W 

F 

C 

Grade. 

W 

F 

C 

(A) 

1,910 

1,002 

185 

(C) 

417 

602 

143 

1,163 

848 

161 

(C(GO 

334 

557 

158 

(C)' 

417 

602 

143 

(C,Gi) 

278 

526 

144 

(CiGi) 

209 

478 

106 

(CtG,) 

139.3 

418 

50 

(A) 

1,910 

1,002 

185 

(G) 

0.451 

61.8 

16 

(A,Gi) 

1,432 

908 

175 

(AsGi) 

1,273 

873 

169 

(Aid) 
(AiG2) 

955 
637 

794 
678 

89 
42 

(HUGO 

10.77 
8.71 

178 
166 

62 

59 

(A,04) 

420 

604 

25 

(E,Gi) 

7.33 

157 

65 

(G) 

0.451 

61.8 

16 

(Eld) 

5.61 

143 

49 

3.89 

127 

33 

(0) 

0.451 

61.8 

16 

(B) 

1,023 

812 

149 

(H.Fl) 

798 

747 

170 

656 

700 

145 

(A) 

1,910 

1,002 

185 

(H!F'I) 

482 

632 

129 

582 

673 

98 

318 

538 

69 

(C)  ' 

417 

602 

143 

(BiF() 

186 

459 

39 

(G) 

0.451 

61.8 

16 

(F) 

1.685 

95.9 

33 

(C) 

417 

602 

143 

(C) 

417 

602 

143 

(CDEFG) 

228 

492 

150 

332 

557 

154 

(D) 

111.5 

388 

127 

(C!EO 

275 

524 

135 

(E) 

10.77 

178 

62 

(CiE,) 

206 

476 

110 

(F) 

1.685 

95.9 

33 

<g,E.) 

137 

406 

91 

(G) 

0.451 

61.8 

16 

88 

358 

75 

(E)' 

10.77 

178 

62 

Natural. 

419 

603 

168 

The  comparisons  of  capacity  with  fineness 
for  series  of  mixtures  of  two  grades  are  illus- 
trated by  figure  63,  where  the  horizontal  scale 
is  that  of  the  proportions  of  fine  and  coarse  in 
the  mixture.  The  capacity  curve  is  identical 
with  the  second  in  figure  55,  and  the  other 
curves  pertain  to  the  same  series  of  (C  G)  mix- 
tures. The  curves  for  capacity  and  linear  fine- 
ness are  strongly  discordant,  capacity  changing 
most  rapidly  with  mixtures  approximately  in 
the  ratio  of  1:1,  and  fineness  changing  most 


rapidly  when  the  proportion  of  the  finer  grade 
is  minute.  The  graph  of  bulk  fineness  is  a 
straight  line  and  betrays  no  sympathy  with  the 
sigmoid  curve  of  capacity,  though  somewhat 
less  discordant  than  the  curve  of  linear  fineness. 
It  is  quite  evident  that  the  peculiar  relation 
of  capacity  to  the  pioportions  of  a  binary  mix- 
ture is  not  to  be  either  accounted  for  or  formu- 
lated as  a  relation  of  fineness,  and  we  have  just 
seen  that  it  can  not  be  formulated  in  terms  of 
the  percentage  of  voids.  The  elimination  of 
those  two  associated  factors  leaves  it — so  far 
as  our  recognized  alternatives  are  concerned — 
to  be  ascribed  wholly  to  modifications  of  the 
texture  of  the  channel  bed  and  the  consequent 
modifications  of  the  mode  of  transportation, 
and  to  these  factors  it  is  not  practicable  to  give 
numerical  expicssion. 

RELATION  OF  CAPACITY  TO  FINENESS,  FOR 
NATURAL  GRADES. 

There  is  another  way  of  comparing  the  ca- 
pacities pertaining  to  mixed  grades  with  the 
fineness  of  the  grades,  which  largely  avoids  the 
influence  of  changing  mode  of  traction  and 
which  throws  a  side  light  on  the  relation  of 
capacity  to  fineness  in  the  case  of  natural 
grades.  Instead  of  comparing  the  data  for 
different  mixtures  of  the  same  two  simple 
grades,  it  compares  data  from  similar  mixtures 
of  different  pairs  of  simple  grades. 

Figure  64  has  been  compiled  from  data  in 
Table  63.  Its  upper  group  of  five  dots  repre- 
sents the  logarithms  of  capacity  in  relation  to 
the  logarithms  of  linear  fineness,  for  combina- 
tions of  fine  and  coarse  in  the  ratio  of  4:1 
(one  ratio  of  3:1  being  included).  The  line 
drawn  among  them  gives,  by  its  inclination, 
an  estimate  of  74,  the  synthetic  index  of 
capacity  in  relation  to  fineness,  for  a  range  in 
fineness  from  166  to  908,  the  value  being  0.70. 
The  next  group  of  dots  corresponds  to  mix- 
tures of  two  fine  to  one  coarse  and  gives  0.57 
as  a  value  of  the  index.  The  next  group,  dis- 
tinguished by  crosses,  corresponds  to  mixtures 
of  one  part  fine  with  one  of  coarse.  It  includes 
six  points,  but  one  of  these  stands  far  from  the 
line  suggested  by  the  others.  The  line,  as 
drawn,  represents  an  index  value  of  0.62.  In 
the  fourth  group,  distinguished  by  X's  and 
corresponding  to  mixtures  of  one  fine  to  two 
coarse,  the  points  are  so  irregularly  placed 
that  no  line  can  be  drawn;  and  a  fifth  group, 


EXPERIMENTS   WITH    MIXED   GRADES. 


181 


not    reproduced,    is    equally   irregular.     Fine- 
ness seems  to  control  capacity  when  the  finer 


Mixture 


1.8 


(CDEFG)       /'  Natural 


FIGURE  M. — Logarithmic  plots  of  capacity  in  relation  to  linear  fineness, 
for  related  mixtures  of  deVis. 

component  of  the  mixture  is  the  more  im- 
portant, but  other  factors  mask  its  influence 
when  the  coarser  component  dominates. 


In  preparing  the  mixture  (C45D35E12FeG2)  the 
endeavor  was  made  so  to  apportion  the  com- 
ponents as  to  approximate  a  natural  grade. 
To  whatever  extent  that  effort  succeeded,  the 
data  obtained  with  use  of  the  mixture  are  com- 
parable with  those  afterward  obtained  with  a 
natural  grade.  The  capacities  and  finenesses 
of  the  complex  mixture  and  the  natural  grade 
are  represented  by  two  plotted  points  near  the 
bottom  of  figure  64,  and  the  line  drawn 
through  them  gives  0.57  as  a  value  of  the 
index.  Unfortunately  the  range  in  fineness 
covered  by  the  two  is  small — from  492  to  603 — 
so  that  their  determination  of  74  has  little 
weight;  its  close  agreement  with  other  values 
may  be  largely  accidental. 

In  Table  45  mean  values  of  74  based  on  work 
with  the  sieve-separated  grades  are  so  arranged 
as  to  show  their  control  by  slope,  discharge,  and 
trough  width.  On  making  the  indicated  allow- 
ances for  differences  in  condition,  and  compar- 
ing those  values  with  the  values  indicated  by 
figure  64,  it  appears  that  the  latter  are  some- 
what smaller  but  that  the  differences  are  not 
great.  The  index  values  from  data  of  the 
experiments  with  mixtures  range  from  0.57  to 
0.70.  Coordinate  values  from  the  experiments 
with  single,  sieve-separated  grades  would  range 
from  0.70  to  0.90. 


500 
Linear  fineness 


1,000 


FIGURE  65. — Curve  illustrating  the  range  and  distribution  of  finenesses  in  natural  and  artificial  grades  of  debris. 


In  considering  the  bearings  of  these  esti- 
mates there  is  advantage  in  giving  graphic 
expression  to  the  conditions  they  severally 
represent.  When  the  results  of  mechanical 
analyses  of  sands  and  similar  materials  are 
presented  graphically,  the  usual  practice  is  to 
plot  integrated  quantities,  or  proportions,  of 
ingredients  on  a  scale  of  fineness  or  coarseness ; 
but  for  the  present  purpose  it  is  convenient  to 
use  the  quantities  without  integration.  In 
figure  65  the  horizontal  scale  is  that  of  linear 
fineness,  and  the  ordinates  are  relative  quan- 


tities of  components  of  different  finenesses. 
The  curve  ABC  represents  the  composition  of 
the  d4bris  constituting  the  tractional  load  of  a 
river  and  is  based  on  a  sample  of  alluvium 
from  a  river  bed.  The  area  ABCD,  between 
the  curve  and  the  axis  of  fineness,  represents 
the  total  load  and  corresponds  to  the  100  per 
cent  of  integrative  diagrams.  The  range  in 
fineness  is  limited  at  A  by  competent  fineness 
for  traction,  coarser  material  on  the  river  bed 
not  being  moved  by  the  particular  discharge 
to  which  the  diagram  corresponds.  It  is 


182 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


limited  in  the  opposite  direction  by  competent 
fineness  for  suspension,  the  area  GED  repre- 
senting suspendible  material  entangled  with 
the  bed  load.  The  mean  fineness  of  the  trac- 
tional  load  is  marked  at  M. 

In  sorting  debris  for  the  laboratory  experi- 
ments, the  material  of  the  river's  load  was 
divided  by  sieves,  and  this  partition  might  be 
represented  in  the  diagram  by  a  series  of 
vertical  lines — slightly  flexed  to  take  account 
of  the  influence  on  the  separation  of  irregularity 
in  shape  of  particles.  The  shaded  areas  X  and 
Y  may  represent  the  constitution  as  to  fineness 
of  two  of  the  sieve-separated  grades. 

The  experiments  show  that  for  a  narrowly 
limited  grade,  such  as  X  or  Y ,  which  has  the 
same  mean  fineness  as  the  river  alluvium, 
ABOD,  the  capacity  for  traction  is  much  less 
than  for  the  alluvium.  They  indicate  also, 
though  less  decisively,  that  for  a  nearly  equal 
mixture  of  a  fine  grade  with  a  coarse,  both 
being  narrowly  limited,  the  capacity  is  nearly 
the  same  as  for  the  unsorted  alluvium,  pro- 
vided the  mean  fineness  is  the  same.  The 
essential  property  appears  to  be  abundance  of 
both  coarse  and  fine,  and  not  multiplicity  of 
grades  of  fineness. 

Passing  now  from  capacity  to  capacity's 
rate  of  variation — in  respect  to  fineness — or  to 
the  valuation  of  it,  we  find  a  certain  parallel- 
ism. For  the  narrowly  limited  grades  the 
sensitiveness  of  capacity  to  fineness,  as  meas- 
ured by  the  index  i4,  extends  upward  from 
about  0.60;  while  values  of  the  synthetic 
index,  It,  range  from  0.70  to  1.00  or  more. 
As  to  these  values  the  data  are  not  abundant. 
For  mixtures  of  two  narrow  grades  we  have  a 
few  estimates  of  It>  of  which  the  largest  is  0.70; 
and  for  grades  similar  to  ABCD  a  single  weak 
estimate  of  0.57.  If  we  conclude  that  the 
sensitiveness  of  capacity  to  fineness  is  less  for 
natural  grades  than  for  the  narrowly  limited 
grades,  we  must  base  the  inference  almost 
wholly  on  the  data  from  the  mixtures  of  two 
grades,  connecting  the  latter  with  natural 
grades  by  aid  of  the  analogy  outlined  above. 
This  I  am  willing  to  do,  but  at  the  same  time  I 
would  record  my  recognition  of  the  weakness 
of  the  evidence  and  reasoning.  It  is  estimated 
that,  on  the  average,  the  capacity  of  streams 
for  natural  grades  of  debris  varies  with  the 
0.60  to  0.75  power  of  linear  fineness.  This  is 
equivalent  to  saying  that  capacity  varies  with 


the  0.20  to  0.25  power  of  bulk  fineness,  or 
with  the  fifth  or  fourth  root  of  bulk  fineness. 

While  the  curve  in  figure  65  is  based  on  the 
mechanical  analysis  of  material  which  consti- 
tuted the  tractional  load  of  a  river  current, 
there  is  no  reason  to  believe  that  its  form  pre- 
sents a  dominant  type.  Inspection  of  other 
analyses,  in  fact,  suggests  that  such  curves  ex- 
hibit much  variety  and  may  sometimes  even 
present  two  maxima.  The  load  which  a  natu- 
ral current  carries  is  determined  not  only  by 
the  two  limits  of  competence,  but  by  the  char- 
acter of  the  material  within  its  reach.  Neigh- 
boring affluents  of  a  river  may  bring  to  it 
strongly  contrasted  grades  of  debris,  or  their 
tribute  may  at  one  time  be  much  finer  than  at 
another.  Moreover,  a  river  is  not  a  simple  cur- 
rent, but  a  complex  of  currents,  which  vary  in 
competence  and  in  the  character  of  their  loads. 
It  is  true  that,  the  channel  being  considered  as 
a  whole,  its  load  at  one  point  is  essentially  the 
same  as  just  above  or  just  below,  but  the  mode 
of  movement  involves  a  continual  remodeling 
of  the  bed  and  a  sorting  and  re-sorting  of  the 
material.  The  load  at  any  particular  point 
and  time  is  conditioned  by  many  factors  of  the 
complex.  For  this  reason  a  representative 
sample  of  a  river's  load  is  not  easy  to  define  or 
to  collect. 

In  view  of  this  complexity  it  is  difficult  to 
apply  even  a  simple  formula  to  problems  in 
river  engineering,  and  refinement  in  formula- 
tion would  be  of  little  avail.  For  the  same 
reason  it  is  not  practicable  to  derive  a  formula 
directly  from  river  data,  and  the  product  of 
the  laboratory  is  the  best  available,  despite  the 
artificial  simplicity  of  its  conditions. 

DEFINITION    AND    MEASUREMENT    OF    MEAN 
FINENESS. 

The  term  "mean  fineness/'  as  here  used,  is 
not  free  from  the  possibility  of  misapprehen- 
sion. As  the  fineness  of  d6bris  is  a  property 
depending  on  the  size  of  component  particles, 
it  is  not  unnatural  to  think  of  fineness  as  a 
property  of  the  particles — and  there  is,  for  that 
matter,  a  fineness  of  particles.  To  obtain  the 
mean  fineness  of  particles,  one  would  first  de- 
termine the  finenesses  of  the  individual  parti- 
cles, and  then  the  mean  of  those  finenesses. 
The  basal  unit  would  be  the  particle.  In  de- 
riving the  mean  fineness  of  a  body  of  debris 
the  basal  unit  is  some  unit  by  which  quantity 


EXPERIMENTS   WITH    MIXED   GRADES. 


183 


of  debris  is  measured.  It  may  be  a  unit  of 
weight  or  a  unit  of  volume.  In  this  report  a 
body  of  debris  is  conceived  to  be  composed  of 
equal  volume  units,  each  of  which  has  a  deter- 
mined or  determinable  fineness,  and  its  mean 
fineness  is  the  mean  of  the  finenesses  of  the 
volume  units.  In  dealing  with  bulk  fineness 
the  mean  computed  is  the  arithmetical  mean 
of  the  finenesses  of  units.  In  dealing  with 
linear  fineness  the  mean  computed  is  the  cube 
root  of  the  arithmetical  mean  of  the  cubes  of 
the  finenesses  of  units. 

The  intricacy  of  the  definition  of  mean  linear 
fineness  arises  from  the  relation  of  linear  fine- 
ness to  bulk  fineness.  The  fundamental  con- 
cept is  that  of  bulk  fineness,  and  the  definition 
of  linear  fineness  rests  upon  it.  Linear  fineness 
is  essentially  a  derivative  of  bulk  fineness,  and 
mean  linear  fineness  is  an  exactly  similar  de- 
rivative of  mean  bulk  fineness.  To  pass  from 
an  assemblage  of  linear  finenesses  to  their 
mean,  it  is  necessary  to  pass  through  bulk 
fineness,  and  that  passage  involves  cubes  and 
cube  root. 

Bulk  fineness  is  defined  as  the  number  of 
particles  in  a  unit  volume  (1  cubic  foot),  it 
being  assumed  there  are  no  voids.  It  is  the 
reciprocal  of  the  volume  of  the  particle — which 
might  be  called  bulk  coarseness.  There  are 
two  practical  modes  of  measuring  it.  If  the 
specific  gravity  of  the  debris  be  known  (or 
assumed),  measurement  includes  a  weighing 
and  a  counting.  Then,  W  being  the  weight, 
N  the  number  of  particles,  G  the  specific 
gravity,  TP0  the  weight  of  a  cubic  foot  of 
water,  and  Ft  the  bulk  fineness, 


W.GN 

~~ 


.(86) 


If  the  specific  gravity  be  not  known,  measure- 
ment includes  two  weighings  and  a  counting. 
Then,  W  being' the  weight  in  air  and  W,  the 
weight  in  water, 


.(87) 


This  procedure  determines  bulk  fineness 
when  all  particles  have  the  same  volume;  when 
they  are  of  different  volumes  it  determines 
mean  bulk  fineness.  As  a  matter  of  fact,  all 
our  measurements  in  the  laboratory  were  of 
mean  fineness.  It  is  not  possible  by  any 


method  of  sorting  with  which  I  am  acquainted 
to  separate  from  a  natural  alluvium  a  grade 
which  is  really  uniform  in  fineness. 

When  the  mean  fineness  of  a  sample  of 
debris  is  desired,  there  is  no  need  to  separate 
it  into  grades,  because  the  process  for  measur- 
ing the  mean  fineness  of  the  whole  is  identical 
with  that  for  measuring  the  fineness  of  a  grade, 
When  bodies  of  d6bris  of  known  finenesses  are 
mingled,  the  mean  fineness  of  the  mixture  is 
computed  by  a  formula  (85),  which  sums  the 
finenesses  by  unit  volumes  (or  weights)  and 
then  divides  by  the  number  of  unit  volumes. 

Linear  fineness  is  defined  as  the  reciprocal 
of  the  mean  diameter  of  the  particles  of  the 
d6bris.  Like  bulk  fineness,  it  is  treated  as  a 
property  of  the  body  of  debris  and  not  as  a 
property  of  the  particle.  Mean  diameter  is 
defined  as  the  diameter  of  a  sphere  having  the 
same  volume  as  the  particle.  Defined  thus, 
linear  fineness  is  a  function  of  volume  of  parti- 
cle, and  as  bulk  fineness  is  also  a  function  of 
that  volume,  the  two  have  a  fixed  relation: 


F-(5 


.(88) 


Substituting  in   (88)  from   (86)   and   (87),  we 
have 


The  computations  of  fineness  for  this  report 
used  (89)  or  (90)  ;  or,  what  is  equivalent,  they 
first  determined  bulk  fineness  by  (86)  or  (87), 
and  then  derived  linear  fineness  by  (88).  The 
computations  of  mean  linear  fineness  applied 
(88)  to  mean  bulk  fineness. 

It  would  have  been  possible  to  formulate  fine- 
ness in  such  a  way  that  the  definition  of  linear 
fineness  would  be  direct  and  comparatively 
simple,  but  any  such  formulation  would  en- 
counter complexity  in  some  of  its  parts,  pro- 
vided it  established  a  logical  relation  between 
linear  fineness  and  bulk  fineness.  Its  adoption 
would  also  -involve  the  sacrifice  of  simplicity  in 
the  measurement  of  fineness.  Any  system  re- 
quiring the  direct  measurement  of  diameters 
would  be  inferior  for  practical  purposes  to  the 
one  here  used. 

The  subject  of  scales  of  fineness  has  been 
elaborated  because  nearly  all  the  results  as  to 


184 


TRANSPORTATION    OF   DEBRIS   BY   RUNNING   WATER. 


the  control  of  capacity  by  fineness  would  be 
quite  different  if  a  different  scale  were  used. 
If,  for  example,  mean  linear  fineness  had  been 
defined  as  the  arithmetical  mean  of  linear  fine- 
nesses, the  curve  for  linear  fineness  in  figure  63 
would  be  a  straight  line,  while  the  line  for  bulk 
fineness  would  be  a  curve  similar  to  that  shown 
for  linear  fineness  but  turned  through  180°. 

When  the  fineness  of  the  tractional  load  of  a 
stream  is  to  be  determined  by  means  of  a  sam- 
ple of  the  d6bris  const ituting  its  bed,  account 
must  be  taken  of  another  factor.  Omitting 
considerations  affecting  the  selection  of  a  sam- 
ple, which  belong  to  Chapter  XIII,  let  us  as- 
sume that  the  sample  hi  hand  is  representative 
of  the  stream's  tractional  load.  In  addition 
to  the  debris  which  was  carried  along  the  bed, 
it  inevitably  includes  finer  material  which  was 
carried  in  suspension.  Suspended  particles  are 
arrested  along  with  the  coarser  and  form  part 
of  every  stream  deposit.  Once  lodged  in  the 
interstices  of  coarser  particles,  they  are  shel- 
tered from  the  current  and  are  not  again  dis- 
turbed so  long  as  the  coarser  material  remains. 
If  the  deposition  of  the  coarser  debris  is  very 
rapid  the  amount  of  entangled  finer  stuff  may 
be  small,  but  when  deposition  is  slow  the  inter- 
stices act  continuously  as  traps  and  catch  sus- 
pendible  debris  until  they  are  filled.  The  lat- 
ter is  the  usual  condition,  and  the  tractional 
sample  therefore  ordinarily  contains  a  consid- 
erable percentage  of  suspensional  material. 
To  separate  the  two  it  is  necessary  to  draw  an 
arbitrary  line,  for  the  graduation  in  fineness  is 
complete.  As  regards  interstitial  space,  the 
tractional  part  of  the  sample  is  comparable  with 
the  more  complex  mixtures  of  the  laboratory, 
and  its  voids  may  be  estimated  as  25  per  cent 
of  the  whole  space.  The  suspensional  d6bris 
packed  in  these  voids  may  be  assumed  itself 
to  include  25  per  cent  of  voids,  so  that  the  net 
volume  of  its  particles  is  three-fourths  that  of 
the  containing  voids,  or  18.75  per  cent  of  the 
whole  space.  The  net  volume  of  the  trac- 
tional particles  being  75  per  cent  of  the  whole 
space,  the  two  divisions  of  the  sample  bear  the 
relation,  by  net  volume  or  by  weight,  of  75  to 
18.75,  or  of  4  to  1.  This  gives  a  practical  rule 
for  separation.  The  sample  should  be  divided, 
with  aid  of  sieves  and  scales,  into  a  coarser  four- 
fifths  and  a  finer  one-fifth,  and  only  the  coarser 
part  should  be  used  in  estimating  mean  fine- 


ness.    In  figure  65  the  entrapped  suspensional 
material  is  represented  by  the  triangular  area 

ODE. 

SUMMARY. 

The  purpose  of  the  experiments  with  mix- 
tures was  to  bring  the  results  from  work  with 
separate  grades  into  proper  relation  with 
phenomena  of  unsorted  natural  material.  The 
indications  given  by  these  experiments  are  in 
part  direct  and  in  part  conditioned  by  the 
principle  adopted  in  framing  a  scale  of  fineness. 
The  adopted  principle  makes  the  conception  of 
bulk  fineness  fundamental  and  that  of  linear 
fineness  derivative. 

The  capacities  for  traction  observed  in  the 
experiments  with  narrowly  limited  grades  are 
less  than  for  equivalent  grades  with  greater 
diversity  in  fineness.  A  study  of  data  from 
mixtures  of  two  narrow  grades  indicates  that 
the  ratio  of  advantage  for  diversified  debris  is 
from  1.17  to  2.22,  the  mean  of  five  estimates, 
from  different  groups  of  data,  being  1 .66.  Two 
comparisons  of  results  from  highly  diversified 
grades,  with  results  from  nearly  homogeneous 
grades  of  the  same  fineness,  give  as  estimates 
of  the  ratio  of  advantage  1.15  and  1.17.  The 
larger  estimates  were  made  by  an  indirect 
method  but  are  independent  of  the  scale  of 
fineness.  The  smaller  estimates  were  made 
by  a  direct  method  but  involve  the  theory  of 
the  scale  of  fineness.  In  combining  the  two 
groups  of  estimates,  greater  weight  is  assigned 
to  the  smaller,  not  because  they  are  of  recog- 
nized higher  authority,  but  because  the  same 
scale  of  fineness  will  almost  necessarily  be 
used  in  applying  the  results  of  the  investiga- 
tion to  practical  questions.  The  compromise 
value  of  1.2  is  adopted,  as  a  correction  to  be 
applied  to  values  of  capacity  in  Table  12  in 
estimating  capacities  for  diversified  grades  of 
like  fineness. 

The  advantage  of  diversification  appears  to 
arise  largely  from  the  fact  that  the  finer 
particles,  by  filling  spaces  between  the  coarser, 
make  a  smoother  road  for  the  travel  of  the 
coarser,  and  it  is  not  proved  that  a  highly 
diversified  debris  gives  higher  capacity  than 
one  containing  only  two  sizes  of  particles. 

It  is  especially  notable  that  when  fine  ma- 
terial is  added  to  a  previously  homogeneous 
coarse  material  not  only  is  the  total  capacity 
increased,  but  the  capacity  for  the  coarser  part 


EXPERIMENTS   WITH    MIXED   GRADES. 


185 


of  the  load  is  increased,  and  it  may  even  be 
enlarged  several  fold.  The  general  effect  of 
adding  coarse  to  fine  is  to  reduce  the  stream's 
capacity  for  the  fine,  but  under  some  conditions 
there  is  a  slight  increase. 

The  general  relations  of  capacity  to  slope, 
discharge,  and  fineness  (and  presumably  to 
form  ratio  also)  are  the  same  for  natural  and 
other  complex  grades  of  de'bris  as  for  the  sieve- 
sorted  grades  of  the  laboratory,  but  some  of 
the  constants  are  not  quite  the  same. 

The  sensitiveness  of  capacity  to  slope  is  on 
the  average  the  same  for  both  classes  of  d6bris 
grades,  but  the  variation  of  sensitiveness  in 
relation  to  slope,  as  determined  by  the  con- 
stant a,  is  somewhat  less  for  natural  grades. 


As  to  the  relation  of  capacity  to  discharge 
comparison  was  limited  to  a  single  example, 
and  that  suggested  no  modification  of  the 
constants  derived  from  work  with  laboratory 
grades. 

The  sensitiveness  of  capacity  to  fineness  is 
somewhat  less  for  natural  grades  than  for  the 
the  laboratory  grades.  No  values  of  the 
constant  <f>  were  obtained  for  complex  grades, 
and  comparisons  of  sensitiveness  were  made 
only  by  means  of  the  synthetic  index  of  relative 
variation.  The  average  value  of  that  index, 
for  natural  grades  of  de'bris  transported  under 
laboratory  conditions,  is  estimated  at  0.20  to 
0.25  for  bulk  fineness. 


CHAPTER  X.— REVIEW  OF  CONTROLS  OF  CAPACITY. 


INTRODUCTION. 

In  the  preceding  seven  chapters  the  relations 
of  capacity  for  stream  traction  to  a  variety  of 
factors  have  been  examined  one  at  a  time.  It 
is  now  proposed  to  bring  together  some  of  the 
discovered  elements  of  control.  The  experi- 
mental data  thus  far  considered  pertain  to 
straight  channels,  and  the  factors  of  control 
connected  with  bending  channels  have  not  re- 
ceived attention.  Those  factors  must  be  in- 
cluded when  the  attempt  is  made  to  bring 
laboratory  results  into  relation  with  river  phe- 
nomena, but  as  they  constitute  a  category  by 
themselves  it  is  convenient  to  leave  them  out 
of  the  account  in  correlating  the  results  from 
straight -channel  work. 

The  immediate  determinants  of  capacity  are 
(1)  the  velocities  of  the  current  adjacent  to  the 
channel  bed,  (2)  the  widths  of  channel  bed 
through  which  those  velocities  are  effective  in 
moving  debris,  and  (3)  the  mobility  of  the 
debris  constituting  the  bed  and  the  load.  It 
was  not  found  practicable  to  measure  bed  ve- 
locity, but  measurement  was  applied  to  its  two 
chief  determinants,  slope  and  discharge,  and 
also  to  its  ultimate  associate,  mean  velocity, 
and  these  have  been  discussed  separately. 
Width  has  entered  into  the  discussion  chiefly  as 
an  associate  of  depth  in  the  determination  of 
form  ratio.  By  reason  of  these  and  other  inter- 
relations the  six  controls  of  capacity  which  have 
been  discussed — slope,  discharge,  fineness, 
depth,  mean  velocity,  and  form  ratio — are  not 
independent,  and  not  all  should  appear  in  a 
general  equation.  Slope,  discharge,  and  fine- 
ness being  accepted  as  of  primary  importance, 
it  is  feasible  to  add  but  one  of  the  others,  and 
choice  has  been  made  of  form  ratio. 

FORMULATION  BASED  ON  COMPETENCE. 

The  functions  used  in  discussing  the  relations 
of  capacity  to  slope,  discharge,  and  fineness  are 
similar,  and  each  involves  a  conception  of  com- 
petence.    Competence  enters  also  the  theoiy  of 
186 


the  relation  of  capacity  to  form  ratio,  but  it 
enters  in  a  different  way.  It  is  convenient  to 
omit  at  first  the  form-ratio  function  and  con- 
sider together  the  three  which  are  similar. 
They  are: 


(10) 

-(64) 
(75) 


Each  of  these  equations  expresses  the  law  of 
variation  of  capacity  with  respect  to  one  con- 
dition when  the  other  two  conditions  are  con- 
stant, and  in  that  sense  they  are  independent  ; 
but  there  is  a  mutual  dependence  of  parameters 
which  is  of  so  complete  a  character  that  they 
are  essentially  simultaneous.  The  dependence 
of  parameters  is  more  readily  stated  by  means 
of  a  specific  instance  than  in  general  terms. 
In  equation  (10)  6,,  a,  and  n  are  constant  so 
long  as  Q  and  Fhold  the  same  values;  they  do 
not  vary  with  variation  of  S.  But  when  the 
values  of  Q  and  F  are  changed  those  of  llf  a, 
and  n  are  modified.  Through  this  control  of  its 
parameters  the  equation  involves  the  relation 
of  capacity  to  discharge  and  fineness. 

The  coefficient  6,  is  the  value  of  capacity 
when  (S-o)  =  \;la  when  (Q-K)  =  l;  &<  when 
(F—<}>)  =  1.  Replacing  them  by  Z»5,  as  the 
numerical  value  of  capacity  when  (S  —  a)  =  \, 
(Q-K)  =  1,  and  (F-  <£)  =  !,  we  may  combine 
the  three  equations  into 


The  constant  J5  is  not  of  the  same  unit  with 
either  &„  b3,  or  &4.  Its  dimensions,  derived  from 
those  of  the  variables  of  (91),  are  L""'30  M+l  T°~l. 

From  the  experimental  data  have  been  com- 
puted 92  values  of  n,  20  values  of  o,  and  5 
values  of  p.  (See  Tables  15,  32,  and  44.)  All 
these  are  positive.  The  following  statistical 
summary  gives  a  general  idea  of  their  relative 
magnitudes.  Its  figures  are  not  based  on  the 
same  range  of  observational  data;  but  the 


REVIEW    OF   CONTROLS   OF   CAPACITY. 


187 


ranges  for  o  and  p  correspond  approximately 
with  the  middle  part  of  the  range  for  n. 


Number 

Expo- 
nent. 

of  deter- 
mina- 

Mean 
value. 

Range  of 
values. 

tions. 

» 

92 

1.59 

0.  93-2.  37 

0 

20 

1.02 

.  81-1.  24 

P 

5 

.58 

.50-  .62 

It  will  be  recalled  that  while  the  forms  of  the 
equations  involving  a,K,  and  <j>  were  based  on  the 
conception  of  competence,  it  was  not  found 
possible  to  correlate  those  parameters  strictly 
with  competent  slope,  discharge,  and  fineness. 
The  correlations  were  obstructed  by  phenomena 
of  dune  rhythms  and  of  diversified  fineness  and 
could  not  be  completed,  but  the  forms  of  equa- 
tion were  found  to  be  well  adapted  to  the  com- 
bined expression  of  observational  data  above 
the  region  of  competence.  Their  relation  to 
competence  is  not  absolute  but  intimate,  and 
it  is  so  intimate  that  certain  properties  of  the 
parameters  may  properly  be  inferred  from  the 
physical  theory  of  competence. 

When  the  swiftest  velocity  on  the  bed  is 
barely  able  to  move  d6bris,  there  is  a  threefold 
condition  of  competence.  For  the  particular 
discharge  and  fineness,  the  slope  is  competent; 
for  the  particular  slope  and  fineness,  the  dis- 
charge is  competent;  for  the  particular  slope 
and  discharge,  the  fineness  is  competent.  The 
conditions  of  competence  for  the  three  factors 
controlling  capacity  are  thus  not  only  similar 
but  simultaneous  and  coincident.  Neither 
factor  can  sink  alone  to  the  limiting  level  of 
competence,  but  the  three  arrive  together. 
This  is  an  important  principle  and  lies  at  the 
foundation  of  the  systematic  interdependence 
of  parameters  and  variables. 


.(92) 


In  equation  (91)  the  quantities 
(S-a),  (Q-K),  and  (F-&  be- 
come zero  simultaneously.  When 
S  =  a,  then  also  Q  =  K,  and  F=  <f> ; 
and  vice  versa. 


As  capacity  varies  directly  with  (S  —  a), 
(Q  —  K~),  and  (F—<f>),  it  is  also  true  that  each  of 
these  varies  directly  with  capacity.  Any 
change  of  condition  which  affects  capacity 
affects  those  three  quantities  in  the  same  sense. 
For  example,  suppose  discharge  to  be  increased. 


This  not  only  increases  (Q  —  K)  and  thereby 
increases  capacity,  but  it  also  increases  (S  —  a) 
and  (F—<f>).  One  mode  of  expressing  this  fact 
is  to  say  that  capacity  measures  the  remoteness 
of  each  controlling  factor  from  the  initial  status 
of  competence,  and  all  recede  or  approach 
together. 

Let  us  now  make  a  more  definite  assumption, 
that  discharge  is  increased  while  slope  and 
fineness  remain  the  same.  The  resulting  in- 
crease of  (S  —  a),  as  S  is  unchanged,  implies  a 
diminution  of  a;  and  the  increase  of  (F—cf>) 
implies  a  diminution  of  <f>.  That  is,  a  and  <f> 
vary  inversely  with  discharge.  Parallel  reason- 
ing shows  that  a  and  K  vary  inversely  with 
fineness,  and  that  K  and  <j>  vary  inversely  with 
slope. 

These  relations  are  here  developed  deduc- 
tively from  the  theory  of  competence.  They 
have  been  developed  inductively  from  the  ob- 
servational data,  for  equations  (26),  (66),  and 
(77)  include 


No  way  has  been  found  in  which  to  study  the 
exponents  deductively.  The  only  evidences  of 
order  discovered  by  comparison  of  observa- 
tional data  pertain  to  n,  which  has  been  found 
(equation  27)  to  vary  inversely  with  discharge 
and  fineness.  The  question  whether  o  and  p 
follow  similar  trends  could  not  be  answered  by 
the  adjusted  data  because  of  the  cumulative 
effect  of  accidental  errors.  There  is,  however, 
considerable  force  in  analogic  reasoning,  based 
not  only  on  equations  (93),  but  on  other  ele- 
ments of  symmetry  in  the  relations  of  capacity 
to  the  several  factors — elements  to  be  noted 
later.  The  state  of  the  evidence  may  be  ex- 
pressed by 


.(94) 


n  =/,„(& 
[o  =/„($, 
[P  =fy  (S, 


It  is  convenient  to  have  a  name  for  the  group 
of  constants  designated  by  Greek  letters,  and 
as  they  define  the  conditions  of  competence, 
they  may  be  called  competence  constants. 

The  exponent  n  and  the  associated  compe- 
tence constant  a,  as  they  vary  with  Q  and  F 


188 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


and  do  not  vary  with  S,  are  controlled  by  Q 
and  F.  As  they  both  vary  inversely  with  Q 
and  with  F,  it  follows  that  they  vary  directly 
one  with  the  other.  It  is  evidently  true  in 
general  that  each  exponent  varies  directly  with 
the  associated  competence  constant. 

--(95) 


?-/l»(0J 

With  a  constant,  the  variation  of  S  —  a  is 
determined  by  variation  of  S.  Considered  as 
additive,  their  variations  are  identical;  but  if 
we  regard  the  changes  as  ratios,  the  changes  in 

(S  —  a)  are  proportional  to  o_    and  those  in  S  to 

«•.     The  ratio  between  these  fractions,  which  is 

o 

t; ,  is  a  measure  of  the  sensitiveness  of  (S  —  a) 

o  —  a 

to  changes  in  S.  It  is  evident  that  as  S  in- 
creases the  sensitiveness  diminishes.  As  ca- 
pacity varies  with  a  power  of  (S  —  a),  the  sensi- 
tiveness of  capacity  to  slope  becomes  less  as  the 
slope  increases. 

Any  change  in  S  causes,  according  to  (92) 
and  (93),  a  change  of  opposite  character  in  K 
and  o.  When  S  is  increased,  K  and  o  are  re- 
duced. As  the  sensitiveness  of  (Q  — «)  to 

change  in  Q  is  measured  by  Q_K,  it  is  evident 

that  the  reduction  of  K  lessens  the  sensitiveness. 
This  has  the  effect  also  of  lessening  the  sensi- 
tiveness of  capacity  to  discharge ;  and  that  sen- 
sitiveness is  further  lessened  by  the  reduction 
of  o.  Parity  of  reasoning  shows  that  increase 
of  slope  lessens  the  sensitiveness  of  capacity  to 
fineness,  so  that  the  effect  of  increasing  slope 
is  to  reduce  the  sensitiveness  of  capacity  to 
all  three  of  its  controlling  factors.  It  is  evi- 
dent also  that  a  similar  result  would  be  reached 
if  the  analysis  began  by  assuming  an  increase 
of  discharge  or  fineness. 

It  is  a  general  principle  that  any  change  in 
one  of  the  control  factors,  slope,  discharge,  and 
fineness,  causing  capacity  to  increase,  has  the 
effect  also  of  making  capacity  less  sensitive  to 
changes  in  each  and  all  of  the  control  factors; 
and  the  inverse  proposition  is  of  course  equally 
true.  The  statement  being  phrased  to  include 
both,  the  sensitiveness  of  capacity  to  the  three 
controlling  conditions  varies  inversely  with 
capacity. 


The  term  "sensitiveness,"  as  used  in  the  pre- 
ceding paragraphs,  is  equivalent  to  the  more 
specific  "index  to  relative  variation,"  for 
which  the  symbol  i  has  been  used;  and  by 
reference  to  various  studies  of  the  control  of 
the  index  by  conditions  it  may  be  seen  that 
the  entire  scope  of  the  general  principle  just 
stated  has  been  covered  by  essentially  in- 
ductive generalizations.  From  equations  (39), 
(68),  and  (79), 


.(96) 


*<-/»($,&  F) 


This  checking  of  deductive  by  inductive  results 
helps  to  establish  the  second  and  third  equa- 
tions of  (94),  which  were  inferred  from  anal- 
ogies. 

Very  little  is  known  of  the  nature  of  the 
functions  in  (93)  to  (96),  beyond  the  fact 
that  those  of  (95)  are  increasing  and  the 
others  decreasing.  Deductive  reasoning  has 
not  been  successfully  applied,  and  induction 
has  escaped  the  entanglement  of  accidental 
errors  in  only  a  single  instance  and  to  a  limited 
extent.  The  exceptional  instance  is  that 
represented  by  the  first  equation  of  group  (96). 
The  symbols  being  translated  into  words,  that 
equation  reads:  The  index  of  relative  varia- 
tion for  capacity  in  relation  to  slope  varies 
inversely  with  slope,  with  discharge,  and  with 
fineness.  There  are  in  fact  three  distinct 
propositions,  and  each  of  these  might  be 
expressed  by  a  separate  equation.  As  to  the 
first  proposition,  that  the  index  varies  in- 
versely with  slope,  it  was  found,  inductively, 
that  the  rate  at  which  it  varies  with  slope  is 
itself  a  decreasing  function  of  slope  and  also 
of  discharge  and  fineness;  and  knowledge  of 
similar  character  was  gained  as  to  the  second 
and  third  propositions -(pp.  104-108).  Repre- 
senting by  diis,  diig,  and  diiF  the  rates  of  varia- 
tion of  the  index  in  relation  to  slope,  discharge, 
and  fineness,  severally,  we  have 


(97) 


di1F=f3(F) 


These  fragmentary  determinations  are  all  of 
one  tenor,  and  in  view  of  the  remarkable 
symmetries  already  discovered  among  the  ele- 


REVIEW   OF   CONTROLS   OF   CAPACITY. 


189 


.(98) 


ments  of  equation  (91),  they  render  probable 
the  general  proposition : 

The  rate  at  which  capacity  varies 
inversely  with  each  of  the  three 
controlling  conditions,  slope,  dis- 
charge, and  fineness,  itself  varies 
inversely  with  each  of  the  condi- 
tions. 

Returning  to  (92)  and  (93),  we  may  indicate 
certain  corollaries. 

Starting  from  the  status  of  competence,  let 
us  assume  that  slope  is  increased,  with  destruc- 
tion of  the  status,  and  that  the  status  is  re- 
stored by  reducing  discharge.  In  the  restored 
status  IT  is  greater  than  in  the  original,  K  is  less, 
and  <j>  is  unchanged.  It  is  evident  that  the 
nature  of  the  result  does  not  depend  on  the 
particular  assumptions,  and  that  we  may  pass 
to  the  general  proposition: 

When  capacity  is  zero,  the  compe- 
tence constants  are  so  related 

that  a  change  in  any  one  of  them  } (99) 

involves   a  change   of   contrary 
sign  in  some  other. 

Starting  from  a  status  characterized  by  a 
particular  value  of  capacity,  we  may  first 
break  it  by  increasing  slope  and  then  restore  it 
by  decreasing  discharge  (fineness  remaining 
unchanged).  The  first  change  reduces  K  and 
<£;  the  second  increases  a  and  <£.  It  does  not 
appear  whether  the  net  result  for  (j>  involves 
change  in  its  value,  but  if  so  the  change  is 
probably  small  in  relation  to  the  increase  in  a 
and  the  decrease  in  «.  It  is  evident  that  the 
nature  of  the  result  does  not  depend  on  the 
particular  assumption,  and  that  we  may  pass 
to  two  general  propositions,  each  of  which 
includes  (99)  as  a  special  case: 


Under  the  condition  that  capac- 
ity is  constant,  the  competence 
constants  are  so  related  that  a 
change  in  any  one  of  them  in- 
volves a  change  of  contrary  sign 
in  some  other. 

Under  the  condition  that  capacit- 
is  constant,  the  values  of  slope, 
discharge  and  fineness  are  so 
related  that  a  change  in  any  one 
of  them  involves  a  change  of 
contrary  sign  in  some  other. 


..(100) 


(101) 


.(102) 


It  follows  also  that 

Under  the  condition  that  capacity 
is  constant,  the  value  of  each 
controlling  condition  (S,  Q,  or 
F)  is  so  related  to  the  corre- 
sponding competence  constant 
(a,  K,  or  <f>)  that  the  two  vary  in 
same  sense. 


Propositions  (100)  and  (102)  are  deduced 
from  equations  (93).  By  parity  of  reasoning 
equations  (94)  yield  (103)  and  (104),  but  these 
two  propositions  share  whatever  uncertainty 
attaches  to  (94). 


Under  the  condition  that  capacity 
is  constant,  the  exponents  n,  o, 
p  are  so  related  that  a  change  in 
any  one  of  them  involves  a 
change  of  contrary  sign  in  some 
other. 

Under  the  condition  that  capacity 
is  constant,  the  value  of  each 
controlling  condition  (S,  Q,  or 
F)  is  so  related  to  the  corre- 
sponding exponent  (n,  o,  or  p) 
that  the  two  vary  in  the  same 
sense. 


..(103) 


(104) 


As  capacity  can  not  be  increased  under  (91) 
without  increasing  S,  Q,  or  F,  and  as  the 
increase  of  one  of  these  involves  under  (93)  the 
decrease  of  two  competence  constants,  without 
any  change  of  the  third,  it  follows  that  the 
competence  constants,  collectively,  vary  in- 
inversely  with  capacity.  The  same  reasoning, 
if  applied  to  (91)  and  (94),  yields  a  similar 
conclusion  as  to  the  exponents.  To  combine 
the  two  in  a  single  statement: 


The  competence  constants  a,  K, 
and  </>,  taken  as  a  group,  and  the 
exponents  n,  o,  and  p,  taken  as 
a  group,  vary  inversely  with 
capacity. 


I  find  it  not  easy  to  bring  into  combination 
the  laws  of  internal  relation  between  para- 
meters of  a  group  and  the  laws  which  connect 
the  groups  with  capacity;  but  if  these  laws  be 
regarded  as  conditions,  it  is  possible  to  frame 
more  comprehensive  theorems  of  tentative 
character.  Equations  (106)  are  of  this  class 


190 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


and  are  thought  worthy  of  examination,  al- 
though the  data  at  hand  do  not  suffice  for  their 
testing. 


....   (106) 


(91) 


The  equation  under  discussion, 


is  an  expression  of  relation  between  capacity 
and  three  of  its  controls,  namely,  slope  (S), 
discharge  (Q),  and  fineness  (F).  It  involves 
seven  parameters,  of  which  six  are  functions 
of  the  independent  variables,  S,  Q,  and  F.  It 
is  thus  a  bare  framework,  and  the  completion 
of  the  structure  calls  for  the  replacement  of  the 
six  parameters  by  their  values  in  terms  of  the 
variables.  The  laws  contained  in  the  equa- 
tions and  propositions  numbered  (92)  to  (98), 
with  their  corollaries,  (99)  to  (105),  are  con- 
tributions toward  the  completion  of  the  struc- 
ture, but  they  are  largely  of  the  nature  of  re- 
strictions. They  impose  conditions  to  be  sat- 
isfied by  the  perfected  equation. 

Some  of  the  conditions  are  already  embodied 
in  the  form  of  (91),  and  with  reference  to  such 
conditions  it  is  important  that  the  origin  of  that 
form  be  not  overlooked.  The  form  assumes 
that  the  three  competence  constants  are  the 
values  of  the  corresponding  variables  when  ca- 
pacity is  zero,  whereas  their  identification  with 
those  values  is  by  no  means  complete.  The 
definition  and  recognition  of  the  status  of  com- 
petence are  so  obstructed  by  the  complicating 
conditions  of  nonhomogeneous  de'bris  and  dune 
rhythm  that  no  more  can  be  asserted  than  an 
indefinitely  representative  relation.  For  most 
purposes,  however,  we  are  little  concerned  with 
conditions  in  the  immediate  neighborhood  of  the 
competence  limit,  so  that  this  qualification  is  of 
small  practical  moment.  Outside  of  the  neigh- 
borhood of  competence  the  support  of  the  form 
is  empiric  ;  it  has  served  well  as  a  scheme  for  the 
marshaling  of  the  observations.  The  support 
is  qualified,  in  turn,  by  the  fact  that  the  obser- 
vations are  not  of  such  harmony  and  precision 
as  to  discriminate  nicely  among  formulas  of 
adjustment.  In  view  of  these  qualifications, 
the  possibility  has  been  recognized  that  some 
of  the  laws  above  enumerated  might  emanate 
from  the  form  of  the  equation  and  have  no 
other  basis;  and  in  view  of  this  possibility  the 


foundations  of  each  conclusion  have  been  scru- 
tinized. I  believe  that  all  the  inferred  laws, 
from  (92)  to  (105),  are  essentially  inductive. 

It  is  easy  to  understand  that  any  construc- 
tive effort  which  should  hang  all  supplementary 
conditions  on  the  framework  of  (91)  would  re- 
sult in  a  formula  so  unwieldly  as  to  be  useless. 
It  is  a  matter  of  faith  with  me  that  if  our  data 
were  so  precise  as  to  substitute  definite  quanti- 
tative relations  for  the  fascicle  of  trends  and 
indefinite  parallelisms  they  have  actually  fur- 
nished, some  way  would  be  found  leading  from 
complexity  to  simplicity.  I  am  not  without 
hope  that  the  presentation  here  made  may  sug- 
gest to  the  mechanist,  familiar  with  the  aspects 
of  solved  problems  of  similar  difficulty,  a  ra- 
tional theory  under  which  the  data  may  advan- 
tageously be  recombined. 

In  an  effort  to  discover  unities  among  the 
complexities  of  the  capacity  relations,  equation 
(91)  was  given  the  following  form: 


The  three  factors  making  the  second  division 
of  the  second  member,  being  independent  of  the 
units  of  measurement,  seemed  well  adapted  to 
the  expression  of  comprehensive  harmonies,  if 
such  exist. 

The  following  negations  were  demonstrated: 

SO  F 

The  quantities  -,  -,  and  -7  are  not  equal,  nor 

are  the  ratios  between  them  constant. 

The  quantities  -  -  1,  ^  -  1,  and  -,  -  1  are  not 

(f  A.  G) 

equal,  nor  are  the  ratios  between  them  con- 
stant. 

The     quantities     f--l),     (~-l),     and 

/F      V 

(  -7  —  1  )    are  not  equal,  nor  are  the  ratios  be- 

tween them  constant. 

It  was  also  found  that  the  symmetric  factors 
in  equation  (91),  namely,  (S  —  e)",  (Q-K)°,  and 
(F—<f>)P,  are  not  equal,  nor  are  the  ratios  be- 
tween them  constant.  . 

THE  FORM-RATIO  FACTOR. 

In  its  relation  to  form  ratio  capacity  has 
two  zeros,  one  corresponding  to  a  high  ratio, 
the  other  to  a  low.  Each  of  these  corresponds 
also  to  a  competent  bed  velocity,  so  that  into 


REVIEW   OF   CONTROLS   OF   CAPACITY. 


191 


a  perfect  formula  competence  would  enter 
twice.  The  formula  adopted,  however,  ignores 
the  element  of  competence,  chiefly  because  its 
recognition,  which  would  add  a  complication, 
was  not  seen  to  be  of  advantage  for  the  expres- 
sion of  the  control  of  capacity  in  the  more  im- 
portant regions  outside  the  vicinity  of  compe- 
tence. The  accepted  formula  is 


The  quantity  p  is  that  value  of  E  which  corre- 
sponds to  the  maximum  value  of  0  —  the  maxi- 
mum standing  between  the  two  zeros  —  and  62 
is  a  capacity  constant.  The  function  as  a 
whole  qualifies  capacity  by  means  of  a  numer- 
ical factor  and  may  be  combined  with  (91)  by 
multiplication  of  the  factors: 


(109) 


The  coefficient  b,  replacing  65  and  b2,  is  a  quan- 
tity of  the  same  unit  with  66  (see  p.  186),  but 
numerically  independent. 

The  function  now  added  is  of  distinct  type 
from  the  others,  for  instead  of  advancing  by  a 
continuous  law  from  zero  to  infinity  it  first 
rises  to  a  finite  maximum  and  then  returns  to 
zero.  The  first  three  factors  are  harmonious; 
the  fourth  discordant.  At  every  stage  in  the 
investigation  the  discussion  of  the  laboratory 
data  has  been  hampered  by  this  discordance. 
In  order  to  treat  adequately  the  relation  of  ca- 
pacity to  either  slope,  discharge,  or  fineness,  it 
was  necessary  to  isolate  that  relation  by  equaliz- 
ing other  conditions,  and  slope  or  discharge  or 
fineness  could  readily  be  equalized;  but  the 
form-ratio  factor  was  intractable.  By  means 
of  interpolation  it  was  possible  to  assemble 
varied  data  characterized  by  the  same  form 
ratio,  but  that  did  not  meet  the  difficulty.  It 
was  necessary  to  take  account  of  the  relation 
of  the  particular  ratio  to  the  optimum  ratio,  p; 
and  the  value  of  ,0.  varies  with  all  other  condi- 
tions. 

The  sensitiveness  of  capacity  to  form  ratio, 
as  measured  by  the  index  of  relative  variation, 
is  less  than  its  sensitiveness  to  other  conditions. 
The  average  of  48  values  tabulated  in  Chapter 
IV  is  0.24,  while  similar  averages  for  fineness, 
discharge,  and  slope  are  three,  five,  and  seven 


times  as  great.  The  distribution  of  sensitive- 
ness, in  relation  to  values  of  the  independent 
variables,  is  illustrated  by  figure  66,  where  four 
curves  are  plotted,  each  representing  a  particu- 
lar instance,  selected  as  typical.  The  vertical 
scale  is  the  same  for  all ;  and  the  ordinates  rep- 
resent values  of  the  index  of  relative  variation. 
The  horizontal  scale  is  that  of  slope  for  the 
curve  SS,  of  discharge  for  the  curve  QQ,  of 
fineness  for  the  curve  FF,  and  of  form  ratio 
for  the  curve  BpR.  The  vertical  cc  represents 
the  competence  constants  and  is  an  asymptote 
to  three  of  the  curves.  The  horizontal  line  mm 
gives  the  value  of  the  exponent  m  correspond- 
ing to  the  form-ratio  index.  For  values  of  R 
greater  than  p  the  index  is  negative,  but  its 
curve  is  drawn  above  the  zero  line  to  represent 
sensitiveness,  which  is  not  affected  by  sign. 


-q  . 


FIGURE  66.—  Typical  curves  illustrating  the  distribution  of  the  : 
tiveness  of  capacity  for  traction  to  various  controlling  conditions. 
Ordinates  represent  values  of  the  index  of  relative  variation;  abscissas, 
to  four  different  scales,  represent  values  of  slope,  discharge,  linear 
fineness,  and  form  ratio. 

The  two  parameters  of  the  form-ratio  factor 
have  laws  of  variation  similar  to  those  of  the 
other  parameters;  each  varies  inversely  with 
values  of  all  independent  variables  except  its 


own, 


P=f(&,Q,  h 

m=/,  (&,Q,h 


(6D 

-62) 


It  follows  that  each  varies  directly  with  each 
of  the  other  parameters  (a,  K,  <p,  n,  o,  and  p). 

The  relations  of  the  parameters  to  form  ratio 
are  less  simple.  In  all  cases  the  evidence  from 
the  observational  data  is  conflicting,  and  as  the 
several  cases  have  come  up  for  consideration 
the  trend  of  evidence  has  seemed  now  in  one 
direction  and  now  in  another.  Impressions  as 
to  those  trends  are  recorded  in  the  preceding 
chapters,  but  when  assembled  they  fail  to  indi- 
cate any  general  principle.  Recourse  is  there- 


192 


TRANSPORTATION    OF   DEBRIS   BY   RUNNING   WATER. 


fore  had  to  theoretic  considerations  alone,  and 
the  conclusions  reached  can  claim  no  direct 
support  from  the  tabulated  values  of  exponents 
and  competence  constants. 

The  conclusion  (105)  that  the  competence 
constants  collectively  vary  inversely  with  ca- 
pacity is  based  on  the  fact  that  individually 
they  vary  inversely  with  things  which  promote 
capacity,  namely,  slope,  discharge,  and  fine- 
ness. Let  us  now  assume  that  those  three  con- 
ditions remain  constant  and  consider  the  effect 
of  varying  form  ratio.  Initially  let  the  form 
ratio  be  so  small  that  the  bed  velocity  is  com- 
petent; a,  K,  and  <f>  are  severally  equal  to  the 
competent  values  of  8,  Q,  and  F.  Now  change 
width  and  depth  so  as  to  increase  the  form 
ratio  and  capacity  becomes  finite.  That  ca- 
pacity may  be  finite,  a,  K,  and  cj>  must  be  less 
than  S,  Q,  and  F,  and  as  the  latter  have  not 
changed  the  competence  constants  have  been 
reduced  by  the  increase  of  form  ratio.  By 
parity  of  reasoning  it  can  be  shown  that  if  the 
initial  form  ratio  be  so  large  as  to  make  the 
bed  velocity  competent  a  reduction  of  form 
ratio  will  cause  a  reduction  of  a,  K,  and  <£. 
Somewhere  between  the  two  form  ratios  of 
competence  lies  p,  the  form  ratio  of  maximum 
capacity,  and  between  the  same  limits  lie  mini- 
mum values  of  the  competence  constants. 

The  greater  the  capacity  induced  by  adjust- 
ment of  form  ratio,  the  greater  the  reduction 
of  slope,  for  example,  necessary  to  reduce 
capacity  to  zero,  and  as  this  reduction  varies 
directly  with  the  depression  of  a  below  the 
initial  value  of  S,  it  follows  that  the  minimum 
value  of  a  (and  similarly  of  «  and  </>)  coincides 
with  the  maximum  of  capacity. 

The  conclusion  that  the  competence  con- 
stants vary  inversely  with  capacity  is  therefore 
true  for  the  case  in  which  changes  in  capacity 
are  caused  by  changes  in  form  ratio.  It  can 
be  shown  also  that  the  exponents,  n,  o,  and  p, 
follow  the  same  law. 

The  extension  of  this  principle  to  the  domain 
of  form  ratio  gives  assurance  that  the  conclu- 
sions embodied  in  equations  and  propositions 
(99)  to  (108),  conclusions  which  were  reached 
from  phenomena  of  slope,  discharge,  and  fine- 
ness, are  not  vitiated  by  the  traversing  phe- 
nomena of  form  ratio. 

The  function  in  the  form-ratio  factor  of  (109) 
being  characterized  by  a  maximum,  the  varia- 
tions of  parameters  with  respect  to  form  ratio 


are  characterized  by  a  minimum.  This  laia 
may  be  so  combined  with  those  of  (93),  (94) 
(61),  and  (62)  as  to  yield  the  following  sys 
tern  of  equations  for  the  trends  of  changes  in 
parameters  consequent  on  changes  in  the  foia 
independent  variables  of  equation  (109): 


P    = 


n  =f, 
o  =/„ 

P    =/7 

»»=/. 


P,  R) 
,  P,  R) 


..(110) 


In  the  development  of  the  form-ratio  factor 
of  equations  (58)  and  (109),  detailed  in  Chap- 
ter IV,  the  factor  first  appeared  as  (1  -  nE)Rm, 
the  quantity  a  being  a  numerical  coefficient  in- 
troduced to  represent  the  resistance  to  the  cur- 
rent occasioned  by  the  sides  of  the  channel.  It 

was  afterward  shown  that<r  =  -^— -  -  and  that 

m+ 1  p 

form  of  coefficient  was  substituted.  These  re- 
lations show  that  either  a  or  m  varies  in  value 
with  the  character  and  amount  of  the  resist- 
ance by  the  channel  sides;  and,  in  point  of  fact, 
both  do.  Nor  is  that  control  restricted  to  the 
parameters  of  form  ratio.  Lateral  resistance 
affects  also,  and  in  comparable  degree,  the  para- 
meters of  slope,  discharge,  fineness,  and  ca- 
pacity. The  sides  of  the  laboratory  channels 
were  vertical  and  were  of  wood,  planed  and 
painted.  Had  they  been  smoother  or  rougher, 
or  had  they  been  inclined,  the  whole  system  of 
values  given  by  the  experiments  would  have 
been  different.  There  is  no  reason,  however 
to  question  that  they  would  have  yielded  the 
same  qualitative  results. 

DUTY  AND  EFFICIENCY. 

The  discussions  of  duty  and  efficiency  in 
Chapters  III  and  V  give  reason  for  the  belief 
that  the  variations  of  either  quantity  in  rela- 
tion to  controlling  conditions  may  advanta- 
geously be  expressed  by  an  equation  identical 
in  form  with  (109).  Such  an  equation  would 
not  be  interconvertible  with  (109),  nor  would 
an  equation  for  duty  be  the  exact  equivalent 
of  one  for  efficiency.  By  the  aid  of  reasonable 
assumptions  the  parameters  of  either  equation 
might  be  derived  from  the  parameters  of  an- 
other, but  the  results  of  computations  by  the 
several  equations  would  not  be  strictly  com- 
patible. These  discordances  may  be  demon- 
strated as  properties  of  the  algebraic  forms. 


REVIEW   OF   CONTROLS   OF   CAPACITY. 


193 


They  are  also  connected  with  the  fact  that  the 
algebraic  forms  imperfectly  represent  the  physi- 
cal phenomena.  It  is  possible  that  the  physi- 
cal facts  are  closely  represented  by  some  one  of 
the  equations,  in  which  case  the  incompati- 
bilities of  the  other  two  equations  are  wholly 
of  the  nature  of  errors;  but  if  this  be  so,  our 
data  do  not  enable  us  to  indicate  the  truer 
equation. 

The  exponent  in  the  slope  factor  is  greater 
for  the  capacity  and  duty  equations  than  for 
the  efficiency  equation,  the  difference  being  a 
large  fraction  of  unity.  The  exponent  in  the 
discharge  factor  is  greater  for  the  capacity 
equation  than  for  those  of  duty  and  efficiency, 
the  difference  being  a  large  fraction  of  unity. 
The  competence  constant  in  the  slope  factor 
is  smaller  in  the  capacity  and  duty  equations 
than  in  that  for  efficiency,  the  difference  being 
a  small  fraction  of  the  constant.  The  com- 
petence constant  in  the  discharge  factor  is 
smaller  for  the  capacity  equation  than  for  the 
others,  the  difference  being  relatively  large. 
The  interrelations  of  parameters  and  the  rela- 
tions of  parameters  to  independent  variables 
are  qualitatively  the  same  for  the  equations  of 
duty  and  efficiency  as  for  the  equation  of 
capacity. 

THK  FORMULA  OP  1ECHALAS. 

THE    FORMULA. 

The  only  earlier  serious  attempt  to  formulate 
the  transportation  of  debris,  so  far  as  I  am 
informed,  is  that  of  C.  Lechalas,  who  wrote  in 
1871,  under  the  title  "Note  sur  les  rivieres  a 
fond  de  sable."1  His  discussion  makes  use  of 
Dubuat's  experiments  on  competent  velocity 
(1786),  Darcy  and  Bazin's  formulas  for  veloci- 
ties in  conduits  and  rivers  (1878),  and  observa- 
tional data  accumulated  by  the  French  engi- 
neering corps  in  connection  with  projects  for 
the  improvement  of  navigable  rivers.  In  the 
following  abstract  of  the  more  elementary 
part  of  his  discussion  the  symbolic  notation 
and  the  terminology  of  the  present  paper  are 
to  some  extent  substituted  for  those  of  the 
original. 

Postulate  a  stream  of  fixed  width  and  con- 
stant discharge,  traversing  a  bed  of  uniform 
sand,  unlimited  in  quantity  but  without 

1  Annales  dcs  ponts  et  <*>aiiss&s,  M&n.  et  doc. ,5th  ser.,  vol.  t,  pp. 381- 
431, 1871. 

20921°— No.  86 — 14 13 


accessions.  So  long  as  the  velocity  along  the 
bed  exceeds  a  certain  value  the  current  trans- 
ports sand.  Below  that  limit,  the  sand  is 
undisturbed. 

The  discharge,  Q,  and  width,  w,  being 
known,  the  mean  depth,  d,  the  slope,  s,  the 
mean  velocity,  Vn,  and  the  bed  velocity,  Vb, 
are  given  by  the  following  three  equations,  of 
which  (1)  and  (3)  are  from  Darcy  and  Bazin. 
The  constants  are  in  meters. 


(1) 

(2) 
(3) 


The  sand  travels  (1)  by  rolling,  (2)  by  sus- 
pension. A  particle  of  water  impinging  on 
the  bottom  gives  motion  to  a  sand  grain,  the 
motion  having  a  direction  which  depends  on 
the  impact  and  on  the  positions  of  adjoining 
particles,  solid  and  liquid.  The  grain  is  pro- 
jected free  from  the  bottom  or  is  rolled  along 
it,  the  particular  result  depending  on  the 
inclination  and  force  of  the  impact  and  on 
various  conditions  which  affect  the  resistance. 
Suspension  corresponds  especially  to  impacts 
associated  with  high  velocities.  Suspension  is 
rare  below  a  certain  critical  velocity  for  each 
density  and  size  of  sand  grain.  Transporta- 
tion is  slow  at  low  stages  of  a  variable  stream, 
rapid  and  by  suspension  at  high  stages.  The 
grains  describe  trajectories  analogous  to  those 
of  the  water  particles,  but  shorter;  and  there 
are  frequent  returns  to  the  bottom,  as  well  as 
restings  between  excursions.  Larger  grains 
are  lifted  less  high,  or  are  rolled  only,  or  remain 
at  rest.  Small  grains  afford  a  better  hold 
(prise)  in  relation  to  their  weight.  The 
smallest  of  all  are  carried  in  the  body  of  the 
current. 

The  amount  by  which  the  pressure  on  the 
upstream  face  of  a  grain  immersed  in  a  current 
exceeds  the  pressure  on  the  downstream  face 
is  proportional  to  the  square  of  the  velocity. 
Represent  it  by  a  Vb2,  the  coefficient  a  depend- 
ing on  size,  form,  and  position.  For  the  sand 
of  the  Loire,  the  resistance  developed  equals 
a  0.252,  as  that  sand  is  immobile  when  Vb  <  0.25. 
"The  difference  is  equal  to  the  product  of  the 
mass  of  the  grains  by  then-  velocity,  projected 
on  the  same  axis  as  F6 — that  is  to  say,  on  the 
axis  of  the  stream.  This  product,  being  pro- 


194 


TRANSPORTATION    OF    DEBRIS   BY    RUNNING    WATER. 


portional  to  the  discharge  [load]  of  sand,  C, 
may  be  represented  by  the  expression  hC.  We 
have  then : 

0=  £(  Vb2  - 0.252)  =  m(  Fb2  - O.oe)-  -  -  (4) 

The  value  of  the  m  will  be  sought  from  obser- 
vation, which  will  correct  in  a  measure  for  the 
introduction  of  F$  into  the  equation  without 
allowance  for  the  speed  of  the  grains,  etc. 
Subtraction  of  0.06  ceases  when  the  sands  are 
prevented  by  suspension  from  rubbing  on  the 
bottom;  therefore  the  formula  becomes 
C=mVb2  for  velocities  above  a  certain  limit. 
(It  is  readily  understood  that  m,  like  a,  is  only 
approximately  constant.)" 

The  rate  at  which  dunes  advance  has  been 
measured,  in  the  French  rivers,  in  relation  to 
the  velocities  of  the  associated  currents.  It 
rises  with  F6  until  the  critical  velocity  is 
reached,  and  then  drops  as  the  change  is  made 
from  rolling  to  suspension.  The  advance  of 
dunes,  depending  on  the  fall  of  grains  into  the 
eddy  (fig.  10)  when  they  have  been  rolled  to 
the  crest,  is  affected  by  the  introduction  of 
suspension  because  then  only  a  part  of  the 
traveling  grains  are  received  by  the  eddy. 

Observations  made  on  the  Loire  give  as  the 
limiting  bed  velocity  for  transportation  [com- 
petent velocity  for  transportation],  F6<,  =  0.25 
meters  per  second.  According  to  an  engineer 
who  has  discussed  those  observations  [H.  L. 
Partiot  ?],  the  corresponding  surface  velocity, 
V,,  is  equal  to  ^JO.ll;  and  the  formula  for  the 
rate  of  advance  of  the  dunes, 

Rate  of  advance  =  0.00013  (FS2-0.11)  .(5) 

is  good  for  all  values  of  Vs  up  to  1.016.  One 
might  base  on  this  a  formula  for  load  in  rela- 
tion to  surface  velocity,  but  the  formula  would 
be  incomplete  unless  developed  so  as  to  take 
account  of  the  depth;  and  it  is  best  for  the 
present  to  adhere  to  equation  (4),  which  con- 
nects load  with  bed  velocity. 

Lechalas,  however,  for  a  temporary  purpose, 
uses  formulas  of  Darcy  and  Bazin  to  connect 
Fb  with  Vs,  under  certain  assumptions  as  to 
depth,  and  with  their  aid  computes  for  the 
Loire  the  critical  bed  velocity,  Vbcc,  at  which 
suspension  of  the  sands  begins.  Fi)0(,  =  0.55 


meters  per  second.  This  is  the  velocity  cor- 
responding to  Fs  =  1.016,  the  surface  velocity 
which  limits  the  applicability  of  formula  (5). 
The  following  table  contains  the  observa- 
tional data  on  dunes  of  the  Loire  and  compares 
the  observed  rates  of  dune  advance  with  rates 
computed  by  formula  (5). 

TABLE  63a. — Data  on  subaqueous  dunes  of  the  Loire. 


Rate  of  dune  advance. 

Surface  ve- 

Height of 

locity. 

dunes. 

Observed. 

Computed. 

Met./sec. 

Meters. 

Met./sec.XW-* 

Met.lsec.XW-* 

0.58 

0.900 

3.0 

3.0 

.64 

.300 

3.3 

3.9 

.73 

.300 

5.1 

5.5 

.75 

.782 

6.3 

5.9 

.81 

.967 

6.7 

7.1 

.81 

.967 

7.5 

7.1 

.83 

.760 

7.6 

7.5 

1.00 

.953 

10.5 

11.6 

1.016 

.920 

12.4 

12.0 

1.016 

.580 

12.0 

12.0 

1.03 

.487 

6.2 

12.35 

1.05 

.612 

7.0 

12.9 

1.11 

1.198 

5.8 

14.6 

1.13 

.650 

8.7 

15.2 

1.33 

.950 

5.6 

21.6 

In  later  passages  Lechalas  recognizes  the 
variations  of  velocity  in  passing  from  one 
vertical  to  another  of  the  same  stream  section 
and  makes  (4)  the  formula  for  a  division,  one 
unit  wide,  of  the  cross  section.  Thus  modified, 
it  is  applied  in  a  variety  of  ways  to  practical 
engineering  problems  of  the  Loire. 

DISCUSSION. 

Lechalas's  classification  of  transportation 
processes  differs  from  that  adopted  for  our 
work  in  that  he  makes  saltation,  at  least 
verbally,  a  part  of  suspension.  I  am  led,  how- 
ever, by  a  study  of  the  more  detailed  descrip- 
tions of  his  colleague  Partiot,  to  believe  that 
the  line  practically  drawn  between  rolling  and 
suspension  differs  in  small  measure  only  from 
the  line  we  have  drawn  between  traction  and 
suspension. 

The  lower  critical  velocity  of  Lechalas  is  the 
exact  equivalent  of  our  velocity  competent  for 
traction,  and  his  upper  critical  velocity  corre- 
sponds approximately  to  our  velocity  compe- 
tent for  suspension.  The  two  attempts  at 
formulation  likewise  agree  in  giving  promi- 
nence to  the  factor  of  competence.  They 
differ  in  the  mode  of  using  that  factor,  and 
they  are  actuated  by  different  preconcep- 
tions. 


KEVIEW   OF   CONTROLS   OF   CAPACITY. 


195 


In  the  first  sentence  of  the  passage  (pp.  193- 
194)  which  has  been  inclosed  in  quotation  marks 
to  indicate  its  literal  translation,  Lechalas  ap- 
pears to  equate  the  velocity  of  a  particle  of  the 
load  with  the  difference  between  the  forward 
pressure  of  the  current  and  the  resistance  given 
by  the  particle.  Hooker1  suggests  that  the  ac- 
celeration of  the  particle  instead  of  its  velocity  is 
intended ;  but  with  or  without  such  emendation 
the  author's  reasoning  is  obscure  to  me,  for  I 
see  no  necessary  physical  relation  between  the 
number  or  mass  of  debris  particles  moved  and 
the  pressure  of  the  current.  The  load  may  be 
defined  as  the  product  of  the  mass  of  particles 
by  their  average  speed;  and  their  speed,  being 
produced  by  the  pressure  of  the  current,  may 
be  simply  related  to  it,  but  any  relation  of  the 
mass  to  the  pressure  is  necessarily  indirect  and 
presumably  involved. 

Whatever  the  strength  or  weakness  of  the 
postulates  on  which  the  formula  is  based,  the 
manner  in  which  it  incorporates  the  principle 
of  competence  gives  it  a  rough  resemblance  to 
those  we  have  developed,  while  the  char- 
acterization of  its  constant  m  gives  to  it  a 
large  empiric  factor;  and  it  is-  in  order  to 
inquire  whether,  as  an  empiric  formula,  it 
finds  support  in  the  Berkeley  data.  As  the 
Berkeley  observations  do  not  include  bed 
velocities,  the  most  direct  comparison  is 
impracticable;  but  an  indirect  relation  may 
readily  be  established. 

The  difficulty  we  have  found  in  defining  bed 
velocity  may  be  avoided,  for  the  purpose  of 

1  Hooker,  E.  H.,  Am.  Soc.  Civil  Eng.  Trans.,  vol.  36,  p.  256, 1896. 


the    present    comparison,    by    accepting    the 
definition  used  by  Lechalas  in 

-    (3) 


and  by  assuming  depth  to  be  constant.  Ac- 
cording to  the  Chezy  formula  this  assumption 
makes  Vm  approximately  proportional  to 
T/S~,  so  that  loads',  in  (3),  is  proportional  to 
Vm.  It  follows  that  Vb  is  proportional  to  Vm, 
and  this  permits  us  to  substitute  Vm  for  F6  in 
equation  (4)  by  changing  the  constants: 

-l<)  __________    (6) 


This  expression  implies  that  capacity  for  trac- 
tion varies  with  mean  velocity  at  a  rate  which 
diminishes  as  mean  velocity  increases  but  is 
never  so  low  as  that  of  the  second  power  of 
mean  velocity.  The  corresponding  data  from 
our  experiments,  namely,  the  data  for  capacity 
in  relation  to  mean  velocity  under  the  condi- 
tion of  constant  depth,  are  in  accord  with  this, 
except  that  they  indicate  a  limiting  index  of 
relative  variation  somewhat  less  than  2.  In 
Table  51  the  values  of  the  synthetic  index,  Ird, 
range  from  2.03  to  7.86;  and  a  value  of  2.03 
for  the  synthetic  index  implies  smaller  values 
of  the  instantaneous  index.  This  discrepancy 
is  not  important,  and  the  formula  of  Lechalas, 
regarded  as  empirical,  is  probably  adequate 
for  the  discussion  of  a  body  of  observations 
on  capacity  and  velocity.  It  could  not,  how- 
ever, be  used  in  connection  with  the  Berkeley 
data  unless  both  K  and  Tc  (or  m  and  0.06  in 
equation  (4))  were  permitted  to  vary  with 
conditions. 


CHAPTER   XI.— EXPERIMENTS   WITH  CROOKED   CHANNELS. 


EXPERIMENTS. 

In  order  to  study  the  influence  which  bends 
in  the  channel  exert  on  capacity  for  traction,  a 
short  series  of  experiments  were  made  with 
channels  having  angular  bends  and  others  with 
channels  having  curved  bends.  Each  of  these 
channels  had  a  width  of  1  foot  and  was  shaped 
by  means  of  partitions  within  a  trough  1.96 
feet  wide.  (See  fig.  67.)  Above  and  below  the 
bends  were  straight  reaches  of  the  same  width. 
All  the  experiments  were  made  with  de"bris  of 


grade  (C)  and  with  a  discharge  of  0.363  ft.3/sec. 
The  loads  were  measured.  In  some  experi- 
ments the  head  lost  in  the  region  of  the  bends 
was  measured  by  means  of  level  readings  on  the 
water  surface  above  and  below. 

After  each  experiment  the  profile  of  the  bed 
was  determined  by  levelings  at  intervals  of  1 
foot,  and  in  several  cases  the  region  of  the 
bends  was  covered  by  such  levelings  and 
sketches  as  to  make  it  possible  to  construct  a 
contour  map  of  the  bed. 


I    7 


n    ? 


m   T 


iv 


FIGURE  67.— Plans  of  troughs  used  in  experiments  to  show  the  influence  of  bends  on  traction. 


SLOPE  DETERMINATIONS. 

The  profiles  of  the  channel  beds  as  shaped 
by  the  current  were  plotted.  Through  that 
part  of  the  profile  corresponding  to  the  straight 
channel  above  the  bends  was  drawn  a  straight 
line  representing  the  mean  slope  for  that  region, 
and  a  similar  straight  line  was  drawn  below  the 
region  of  the  bends.  From  a  point  on  the  first 
line  near  the  position  of  the  first  bend  to  a  point 
on  the  second  line  corresponding  to  a  position 
several  feet  below  the  last  bend  a  straight  line 
was  drawn,  and  this  was  assumed  to  represent 
the  mean  slope  of  the  channel  in  the  region 
affected  most  by  the  bends.  The  profiles  above 
the  bends  showed  evidence  of  the  rhythms  com- 
monly observed  in  the  straight^channel  experi- 
ments. The  profiles  below  the  bends  showed 
steeper  undulations,  which  were  ascribed  to 
196 


the  influence  of  the  strong  agitation  of  the 
water  in  passing  the  bends.  In  estimating 
the  slopes  in  the  region  of  the  bends  the  dis- 
tance used  was  the  length  of  the  medial  line  of 
the  channel. 

The  observations  for  head  were  made  at 
points  2  feet  and  4  feet  above  the  first  bend 
and  3  feet  and  5  feet  below  the  last  bend,  and 
the  slopes  were  computed  for  the  distances,  on 
the  medial  line,  between  the  points. 

Check  estimates  of  slope  for  straight  chan- 
nels were  obtained  by  interpolation  from  Table 
12,  the  determinations  of  load  being  used  as 
arguments. 

The  data  are  assembled  in  Table  64,  where 
the  stronger  and  weaker  determinations  of  the 
slope  of  the  debris  surface  are  severally  indi- 
cated by  the  letters  a  and  &.  The  measure- 
ments of  water  slope  are  thought  to  be  coordi- 


EXPERIMENTS   WITH    CROOKED   CHANNELS. 


197 


nate  in  value  with  those  of  debris  slope.  The 
slopes  computed  from  load  measurements  are 
probably  of  less  weight. 

TABLE  64. — Comparison  of  slopes  required  for  straight  and 
crooked  channels,  respectively,  under  identical  conditions 
of  discharge,  fineness,  width,  and  load. 

[DeT>ris  of  grade  (C);  discharge,  0.363  fU/sec.;  width,  1  foot.] 


Shape  of 
crooked 
channel. 

(See  fig. 
67.) 

Load. 

Slope  in  straight 
channel. 

Slope  in  crooked 
channel. 

Computed 
from  load 
by  Table 
12. 

Profile  of 
de'bris 
above  first 
bend. 

Average 
slope  of 
de'bris. 

Average 
slope  of 
water 
surface. 

I  

Gm.iiec. 
19 

75 
56 

60 
61 

60 
60 

63 
63 
66 
64 

64 

Pmccnt. 
0.42 
.92 

.77 

.80 
.82 

.80 
.80 

.83 
.83 

.85 
.84 

.84 

Per  cent. 
0.  30  6 
.896 
.746 

.70  a 
.946 

.76  a 
.86  a 

.726 
.626 
.876 
.87  a 
.86  a 

Per  cent. 
0.456 
.89  o 
.80o 

.816 
1.15  a 

.80a 
.856 

.836 

.806 
.81  0 
.77o 
.SO  (i 

Per  cent. 
0.46 
.93 

.78 

.72 

II  
Ill  
IV  

V  

NOTE.— Values  marked  n  are  given  greater  weight  than  those  marked  6. 

FORMS  AND  SLOPES. 

The  combinations  of  bends  in  the  experiment 
channels  are  shown  in  figure  67.  In  three  of 
the  channels  the  bends  were  angular;  in  two 
curved.  In  channel  I  the  angle  of  deflection 
was  10.5°;  there  'was  a  single  group  of  four 
bends,  returning  the  course  to  its  original  direc- 
tion ;  and  the  short  reaches  were  approximately 
5  feet  long.  In  channel  II  the  arrangement 
was  the  same,  -with  reaches  of  about  2.5  feet 
and  deflection  angles  of  21.5°.  In  channel  III 
were  three  groups  of  four  deflections  each,  the 
angles  being  of  40.9°  and  the  length  of  reach 
about  1.4  feet.  Channel  IV  had  the  same  pro- 
portions as  No.  II,  with  the  substitution  of 
curves  for  angles.  The  radius  of  curvature  for 
the  medial  line  was  6.55  feet.  Channel  V  con- 
tained two  groups  of  curves,  each  similar  to  the 
group  in  No.  IV. 

From  the  data  (Table  64)  connected  with 
channels  I,  II,  and  III,  it  appears  that  with 
angular  bends  a  greater  slope  is  necessary  to 
transport  the  load  than  when  the  channel  is 
straight — that  is,  the  capacity  is  reduced  by 
angular  bends.  The  reduction  is  greatest  when 
the  angle  of  deflection  is  greatest,  and  it  is  so 


small  for  an  angle  of  10°  as  to  bave  doubt 
whether  it  might  not  disappear  altogether  with 
a  somewhat  smaller  angle. 

The  single  group  of  curves  (IV)  appears  to 
reduce  capacity  slightly  (increase  of  slope  for 
same  load):  but  the  double  group  (V)  gives 
slopes  indicating  an  increase  of  capacity. 

As  the  bends  of  alluvial  streams  are  curved, 
the  curved  experiment  channels  may  be  as- 
sumed to  represent  them  better  than  do  the 
angular  channels,  and  it  is  possible  that  mean- 
dering channels  have  a  greater  capacity  for 
traction  than  straight  channels  of  the  same 
length.  There  are,  however,  certain  elements 
of  incompleteness  in  the  representation  which 
make  definite  inference  hazardous.  The  course 
of  a  stream  which  shapes  its  own  channel 
through  an  alluvial  plain  is  made  up  of  bends 
and  reaches.  In  passing  from  reach  to  bend 
there  is  a  gradual  increase  of  curvature  until 
the  radius  of  curvature,  for  the  medial  line,  is 
between  twice  and  three  times  the  width  of  the 
channel,  and  the  change  from  bend  to  reach  is 
also  gradual.  The  forms  are  automatically 
adjusted  to  the  system  of  accelerations  and 
velocities  within  the  current.  The  angular 
change  of  direction  in  the  bend  may  be  one  of 
a  few  degrees  only,  but  in  meandering  streams 
it  is  commonly  from  90°  to  180°.  In  the  arti- 
ficial channels  all  curves  were  circular  arcs  with 
a  radius  of  6.55  times  the  channel  width;  there 
was  no  graduation  in  the  radial  acceleration 
due  to  deflection;  the  change  from  right-hand 
deflection  to  left-hand  deflection  was  abrupt, 
without  the  intervention  of  a  reach,  and  the 
changes  of  direction  were  through  angles  of 
21.5°  and  43°.  That  such  differences  are  com- 
petent to  affect  transportation  to  a  material 
extent  is  indicated  by  the  relations  of  deeps 
and  shoals  (crossings)  to  bends.  Fargue,1  from 
a  discussion  of  an  artificially  adjusted  portion 
of  the  Garonne,  reached  the  conclusion  that  the 
distance  downstream  from  the  apex  of  a  curve 
to  the  deepest  point  of  the  associated  deep  and 
the  distance  downstream  from  a  point  of  inflec- 
tion to  the  associated  crossing  are  each  nor- 
mally one-fourth  the  length  of  a  stream  unit — 
defined  as  the  portion  between  two  points  of 
inflection.  In  our  experiments  (see  fig.  68) 
each  of  these  distances  is  one-half  the  length 
of  the  stream  unit  instead  of  one-fourth. 

1  Fargue,  L.,  La  forme  du  lit  des  rivieres  a  fond  mobae,  Paris,  1908. 


198 


TRANSPORTATION    OF    DEBRIS  BY   RUNNING   WATER. 


rUour  interval  O.O2 foot 
Width  oftroufffiJfoot 


FIGURE  68. — Contoured  plot  of  a  stream  bed,  as  shaped  by  a  current. 


In  view  of  these  facts,  the  apparent  results 
of  the  experiments  with  crooked  channels  must 
be  received  with  caution,  and  probably  nothing 
more  should  be  claimed  for  them  than  a  general 
indication  that  the  capacity  of  a  moderately 
bent  channel  does  not  differ  greatly  from  that 
of  a  straight  channel. 

FEATURES  CAUSED  BY  CURVATURE. 

Incidentally  the  experiments  illustrated  sev- 
eral consequences  of  curvature  in  addition  to 
the  influence  on  slope  and  capacity.  At  each 
turn  the  swiftest  part  of  the  current  was  thrown 
to  the  outer  or  concave  side  of  the  channel,  and 
the  slower  parts  moved  toward  the  opposite 
side,  the  transfers  giving  to  the  current  as  a 


whole  a  twisting  motion.  The  action  on  the 
debris  became  exceptionally  strong  near  the 
outer  side  and  exceptionally  weak  near  the 
inner.  A  result  of  the  strong  action  was  that 
part  of  the  load  was  thrown  upward,  so  as  to 
be  temporarily  suspended,  and  a  result  of  the 
diversity  of  velocity  was  the  maintenance  of 
deep  places  near  the  outer  wall  and  of  shoals 
near  the  inner.  Associated  with  the  twisting 
motion  were  many  whirls  or  eddies;  and  the 
general  obliquity  of  motion  had  the  effect  of 
reducing  the  mean  velocity  in  the  direction  of 
the  general  flow.  The  reduction  of  mean 
velocity  was  recorded  in  an  increase  of  mean 
depth,  which  amounted,  in  the  average  of  all 
examples,  to  7  per  cent  and  ranged  from  2  to 
14  per  cent. 


CHAPTER   XII.— FLUME   TRACTION. 


THE  OBSERVATIONS. 

SCOPE. 

That  which  distinguishes  flume  traction  from 
stream  traction  is  the  fixity  of  the  channel  bed. 
In  stream  traction  the  shapes  of  the  bed  are 
adjusted  to  the  rhythms  of  the  mode  of  trans- 
portation, and  its  texture  is  that  of  the  debris 
in  transit.  In  flume  traction  the  bed  is  unre- 
sponsive, but  its  texture,  being  independently 
determined,  has  an  important  influence  on  the 
mode  of  transportation.  The  experiments 
were  arranged  to  determine  the  influences  of 
different  textures  of  bed  on  mode  of  traction 
and  capacity  for  traction  and  were  otherwise 
varied  in  respect  to  slope,  width,  discharge, 
and  the  character  of  debris  transported. 

GRADES    OF   DEBRIS. 

The  material  transported  in  the  experiments 
included  most  of  the  grades  already  described 
(see  Table  1  and  Plate  II)  and  also  several 
mixtures  not  previously  mentioned.  In  order 
conveniently  to  show  the  relations  of  the  mix- 
tures to  their  components,  all  the  grades  of 
debris  used  in  the  flume  experiments  are  listed 
below  in  Table  65,  the  data  of  Table  1  being 
repeated  so  far  as  necessary.  The  elements  of 
the  table  are  defined  at  page  21.  The  material 
of  the  coarse  grades  (I)  and  (J)  differs  from 
that  of  the  finer,  being  about  2  per  cent  less 
dense.  Its  particles  also  are  somewhat  less 
thoroughly  rounded,  their  journey  from  the 
parent  rock  bed  having  been  short. 

TABLE  65. — Grades  of  debris . 


Grade  name. 

Sieves 
used  in 
separation 
(meshes 
to  1  inch). 

D. 

Mean  di- 
ameter of 
particles 
(foot). 

F. 

Number 
of  par- 
ticles to 
linear 
foot. 

Ft. 

Number  of 
particles  to 
cubic  foot. 

(B)  . 

40-50 

0.00123 

812 

1.023.000,000 

(C)... 

30-40 

.00168 

602 

417,000,000 

(E) 

10-20 

.00561 

178 

10,770,000 

(G)... 

4-6 

.0162 

61.8 

451,000 

H) 

3-4 

.0230 

43.4 

156,000 

i)..:: 

1-2 

.0547 

18.3 

11,900 

(j) 

J-l 

.110 

9.1 

1,440 

(EjGi)... 

.00698 

143.2 

5,610,000 

(EjHiIj) 

.00706 

141.7 

5,430,000 

(EjHjIjJj).. 

.00836 

119.6 

3,266,000 

APPARATUS    AND   METHODS. 

The  experiment  trough,  a  modification  of 
that  represented  in  the  frontispiece,  was  60 
feet  long  and  1.91  feet  wide,  with  vertical 
sides.  The  sides  and  bottom  were  of  wood, 
planed  and  painted.  For  a  portion  of  the 
experiments  the  bottom  was  covered  by  a 
false  bottom,  specially  prepared  to  present  a 
definite  character  of  roughness,  the  sides  re- 
maining smooth.  The  trough  was  so  arranged 
that  it  could  be  given  various  determinate 
slopes  up  to  3  per  cent,  and  by  means  of  an 
inclined  false  bottom  a  slope  of  4.5  per  cent 
was  made.  By  means  of  a  partition  the  width 
was  reduced,  for  the  greater  part  of  the  work, 
to  1.00  foot. 

The  width  of  trough  at  the  outfall  end  was 
regulated  by  a  contractor,  as  described  on 
page  25.  The  debris  was  delivered  at  the 
outfall  to  a  settling  tank,  which  had  two 
divisions;  and  a  deflecting  apparatus  was  so 
arranged  that  the  delivery  could  be  instanta- 
neously diverted  from  one  division  to  the  other. 

Above  the  trough  near  its  head  was  a  sloping 
platform  on  which  measured  units  of  debris 
were  dumped  at  regularintervals,  determined  by 
a  watch,  and  from  which  the  debris  was  fed  to 
the  current  by  hand,  with  the  aid  of  a  scraper. 
The  rate  of  £eed  was  modified  by  changing  the 
interval  between  dumpings,  and  by  successive 
trials  it  was  adjusted  to  the  capacity  of  the 
current. 

In  accelerating  the  debris,  as  it  fell  into  the 
water,  the  current  was  retarded,  so  that  close 
to  the  feeding  station  it  was  slower  than  else- 
where. When  the  load  was  approximately 
adjusted  to  the  general  capacity  of  the  current 
it  constituted  an  overload  in  this  particular 
tract,  with  the  result  that  a  portion  was  de- 
posited. The  load  would  then  traverse  an 
upper  division  of  its  course  on  a  bed  of  debris, 
while  in  the  lower  and  principal  division  it  was 
in  direct  contact  witn  the  bottom  of  the  trough. 
A  tendency  of  the  stream  of  debris  to  clog  near 
the  upper  end  of  the  trough,  although  moving 
freely  beyond,  was  the  ordinary  criterion  of  the 

199 


200 


TRANSPORTATION    OF   DEBRIS   BY   RUNNING   WATER. 


proper  adjustment  of  the  feed,  and  when  this 
condition  existed  the  load  delivered  at  the  out- 
fall was  assumed  to  represent  the  capacity  of 
the  stream.  The  outfall  was  then  directed  for 
a  measured  interval  of  time  to  a  reserved 
division  of  the  settling  tank,  and  the  d6bris 
thus  separately  received  was  weighed. 

Besides  the  deposit  connected  with  the  feed- 
ing of  debris  there  were  transitory  deposits  of 
a  rhythmic  character,  as  described  later. 

Five  characters  of  channel  bed  were  used, 
namely,  a  planed  and  painted  wood  surface;  a 
rough-sawn,  unplaned  wood  surface;  a  surface 
of  wood  blocks,  with  grain  vertical;  a  pavement 
of  sand  grains,  set  in  cement;  and  a  pavement 
of  pebbles.  (See  PI.  III.) 

PROCESSES    OF    FLUME    TRACTION. 
MOVEMENT  OF  INDIVIDUAL  PARTICLES. 

Flume  traction  differs  from  stream  traction 
in  its  extensive  substitution  of  rolling  for  sal- 
tation and  in  the  important  place  it  gives  to 
sliding.  The  relative  importance  of  these 
modes  of  particle  movement  is  determined  (1) 
by  the  texture  of  the  bed  surface  in  relation  to 
the  size  of  the  particle,  (2)  by  the  velocity  of 
the  water  in  relation  to  the  size  of  the  particle, 
and  (3)  by  the  shape  of  the  particle. 

In  stream  traction  the  order  of  roughness  of 
the  bed  is  given  by  the  fineness  or  coarseness  of 
the  material  of  the  load,  and  this  fact  deter- 
mines saltation  as  the  dominant  process.  In 
flume  traction  the  bed  may  be  much  smoother. 
On  a  smooth  stream  bed  any  particle  with  a 
broad  facet  is  apt  to  slide,  and  a  well-rounded 
particle  to  roll.  Rolling  is  determined  (rather 
than  sliding)  not  only  by  the  fact  that  the  pro- 
pulsive force  of  the  current  and  the  resistance 
given  by  the  bed  constitute  a  couple,  but  also 
by  the  fact  that  the  current  applies  a  greater 
force  to  the  upper  part  of  the  particle  than  to 
the  lower.  The  less  smooth  the  bed  surface, 
the  greater  its  resistance  and  the  more  effective 
the  couple  in  causing  the  particle  to  roll.  With 
any  particular  texture  of  bed,  the  sizes  of  par- 
ticles may  determine  their  modes  of  progress 
the  largest  sliding,  those  of  smaller  size  rolling, 
and  the  smallest  leaping.  Increase  of  velocity 
tends  to  increase  saltation  at  the  expense  of 
rolling  and  to  increase  rolling  at  the  expense 
of  sliding. 

A  particle  rolled  slowly  is  in  continuous  con- 
tact with  the  bed.  A  round  particle  may  roll 


rapidly  on  a  smooth  bed  without  parting  from 
it.  Roughness  of  the  bed  causes  changes  of 
direction  in  the  vertical  plane,  and  such  changes 
combined  with  high  velocity  cause  leaps.  If 
the  particle  is  not  round  its  rolling  involves 
rise  and  fall  of  the  center  of  mass,  and  such 
changes  combined  with  high  velocity  cause 
leaps.  Shape  of  particle  may  thus  be  a  deter- 
minant between  saltation  and  rolling,  as  well 
as  between  rolling  and  sliding. 

A  flattish  particle,  which  may  either  slide  or 
roll,  travels  faster  when  rolling.  This  is  due 
partly  to  the  fact  that  when  it  rolls  it  rolls  on 
edge  and  thus  projects  farther  into  the  current, 
and  partly  to  the  fact  that  the  resistance  at 
contact  with  the  bed  is  greater  for  sliding  than 
for  rolling.  It  is  also  true  that  rolling  par- 
ticles as  a  class  outstrip  sliding  particles  as  a 
class,  the  difference  in  speed  being  marked. 

For  particles  of  similar  size  those  which  domi- 
nantly  roll  outstrip  those  which  dominantly 
leap.  This  is  part  of  a  more  general  fact  that 
the  better-rounded  particles  travel  faster  than 
the  more  angular.  In  the  traction  of  mixed 
debris,  where  rolling  is  characteristic  of  larger 
particles  and  saltation  of  smaller,  the  larger 
travel  faster  than  the  smaller.  There  are  thus 
two  important  ways  in  which  rolling  gives 
greater  speed  than  saltation.  It  was  not 
learned  whether  a  particle  which  alternately 
rolls  and  leaps  travels  faster  in  one  way  than 
in  the  other. 

A  suspended  particle,  having  the  same  speed 
as  the  water,  outstrips  all  others.  It  is  there- 
fore possible  that  as  saltation  approaches  the 
borderland  of  suspension  its  speed  exceeds  that 
of  rolling. 

When  samples  of  different  grades  are  fed  tc 
the  same  current  in  succession,  it  is  found  that 
the  coarser  travel  the  faster,  whatever  the  mode 
of  progression  (except  suspension).  In  the  fol- 
lowing record  of  experiments  the  speeds  of 
grade  (J)  constitute  an  apparent  exception,  but 
their  slowness  is  ascribed  to  the  fact  that  the 
particles  of  that  grade  were  relatively  angular. 

TABLE  66. — Relative  speeds  of  coarse  and  fine  debris  in  flume 
traction. 


Depth  of 
water. 

Mean  veloc- 
ity of  water. 

Average  speed  of  d<5bris  particles. 

Grade  (E).  i  Grade  (H). 

Grade  (J). 

Foot. 
0.127 
.182 
.110 
.182 

Ft.lsec. 
4.35 
3.04 
2.52 
1.S2 

Ft.lsec. 
3.2 
2.3 
1.85 
.70 

Ft./sec. 
3.9 
2.4 
2.0 

.85 

Ft.lsec. 
3.8 

2.0 

U.  S.   GEOLOGICAL  SURVEY 


PROFESSIONAL  PAPER  86     PLATE   III 


nut 


ROUGH    SURFACES    USED    IN     EXPERIMENTS    ON     FLUME    TRACTION. 


FLUME    TRACTION. 


201 


It  may  be  observed  in  passing  that  the  re- 
corded speed  of  the  particles  is  on  the  average 
75  per  cent  of  the  mean  velocity  of  the  water, 
the  ratio  being  gi  eater  as  the  velocity  is  greater 
and  as  the  depth  is  less.  A  higher  ratio  would 
of  course  be  found  if  the  speed  of  particles 
were  to  be  compared  with  that  of  the  lower  part 
of  the  current.  The  percentage  might  be  less 
than  75  if  the  stream  were  fully  loaded. 

In  watching  the  traction  of  mixed  debris  it 
was  observed,  as  already  mentioned,  that  the 
larger  particles  traveled  faster  than  the  smaller. 
As  it  is  difficult  in  such  an  observation  to  avoid 
giving  attention  largely  to  the  more  active  par- 
ticles, the  observation  applies  especially  to  those 
which  roll,  but  there  is  probably  a  similar  con- 
trast between  the  speeds  of  less  active  parti- 
cles also.  In  an  experiment  with  glass  balls  of 
different  sizes  it  was  found  that  the  larger  were 
rolled  by  the  current  somewhat  faster  than  the 
smaller. 

In  attempting  to  understand  the  more  rapid 
propulsion  of  the  larger  particles  it  is  natural 
to  compare  the  phenomenon  with  the  more 
familiar  fact  that  in  the  absence  of  water  a 
large  stone  or  ball  will  descend  an  incline  with 
greater  speed  than  a  small  one,  but  there  is  an 
important  difference  between  the  two  cases. 
The  object  descending  a  dry  incline  is  impelled 
by  gravity,  acting  directly,  and  part  of  the 
resistance  comes  from  the  fluid  in  which  it  is 
immersed,  whereas  the  rolling  debris  pebble  is 
impelled  chiefly  by  the  moving  fluid  which 
surrounds  it.  The  advantage  in  speed  accruing 
to  the  larger  pebble  in  flume  traction  is  prob- 
ably connected  with  the  fact  that  the  velocities 
of  the  current  increase  from  the  bottom 
upward.  If  the  velocities  were  all  the  same 
the  pressures  applied  to  similar  pebbles  of 
different  diameter  would  be  proportional  to 
their  sectional  areas,  or  to  the  squares  of  their 
diameters;  but  because  of  the  gradation  of 
velocities  the  increase  of  pressure  in  conse- 
quence of  increase  of  diameter  is  more  rapid 
than  the  increase  of  the  square  of  the  diameter. 
The  chief  resistance  to  the  forward  movement 
of  the  pebble  is  engendered  at  its  contact  with 
the  bed  and  is  of  the  nature  of  rolling  resistance. 
For  uniform  speed,  the  rolling  resistance  of  a 
wheel  is  proportional  to  its  downward  pressure 
divided  by  its  diameter;  and  since  in  the  case 
of  the  pebble  the  pressure  is  proportional  to 
the  cube  of  the  diameter,  the  rolling  resistance 
is  proportional  to  the  square.  Increase  in  size 
of  the  rolling  pebble  is  thus  accompanied  by 


increase  of  both  propulsive  force  and  rolling 
resistance,  the  increment  to  propulsive  force 
being  somewhat  the  larger;  and  the  equality  of 
force  and  resistance  is  restored  by  an  increase 
of  speed,  which  has  the  effect  of  reducing  the 
propulsive  force  and  increasing  the  resistance.1 
The  reduction  of  propulsive  force  is  connected 
with  the  fact  that  that  force  is  determined  at 
each  instant  by  the  velocities  of  the  water,  not 
as  referred  to  the  fixed  bed,  but  as  referred  to 
the  moving  pebble. 

The  analysis  of  forces  might  be  further  de- 
veloped, but  the  foregoing  brief  outline  serves 
to  indicate  a  mechanical  principle  underlying 
the  observed  fact  that  a  current  rolls  large 
particles  more  swiftly  than  small.  The  princi- 
ple is  of  fundamental  importance  in  account- 
ing for  certain  contrasts  between  the  laws  of 
flume  traction  and  those  of  stream  traction. 

The  discrimination  of  traction  and  suspen- 
sion, usually  easy  in  the  experiments  with 
stream  transportation,  was  difficult  in  the 
flume  work.  The  zone  of  saltation  grew 
deeper  with  progressive  increase  of  slope  until 
it  occupied  the  whole  depth  of  the  stream. 
Further  increase  of  slope,  with  increase  of  load, 
made  the  cloud  of  particles  denser,  but  there 
seemed  no  way  of  telling  when  the  condition 
became  that  of  a  flowing  mixture  of  water 
and  sand. 

COLLECTIVE  MOVEMENT. 

To  the  general  fact  that  in  flume  traction 
the  particles  of  the  load  either  roll  or  slide  in 
continuous  contact  with  the  fixed  bed  or  else 
skip  from  point  to  point  along  it,  there  are  two 
noteworthy  exceptions. 

When  the  transported  debris  includes  par- 
ticles which  are  small  compared  to  the  projec- 
tions constituting  the  roughness  of  the  bed, 
some  of  the  debris  finds  lodgment  among  the 
projections.  The  roughness  of  the  bed  is  thus 
diminished  and  the  process  of  transportation 
is  modified.  The  bed  comes  to  be  constituted 
in  part  of  the  fixed  summits  of  projections  and 
in  part  of  mobile  debris,  and  the  process  be- 
comes a  blending  of  flume  traction  proper  and 
stream  traction.  When  such  conditions  existed 
in  the  experiments  it  was  found  that  the  capacity 
was  essentially  that  due  to  stream  traction. 

i  The  formula  given  by  W.  J.  M.  Rankine,  in  his  "  Applied  mechanics  " 
and  elsewhere,  for  the  rolling  resistance  of  a  wheel  is  R— -|a+6(t)— 3.28)} 

where  R  is  resistance,  Q  gross  load,  r  radius  of  wheel,  v  velocity  in  ft. /sec., 
and  a  and  b  constants.  The  experimental  values  of  a  and  b  are  such  as  to 
give  velocity  only  a  moderate  influence  on  the  resistance. 


202 


TRANSPORTATION    OF    DEBRIS    BY   RUNNING    WATER. 


The  second  exception  is  a  phenomenon  of 
rhythm.  It  is  probable  that  the  rate  of  flume 
traction  is  always  affected  by  rhythm,  just  as 
is  that  of  stream  traction,  and  the  rhythm  in 
load  implies  a  rhythm  in  efficiency  of  current. 
Under  certain  conditions,  of  which  the  most 
important  is  small  slope,  the  rhythm  is  mani- 
fested by  the  making  of  a  local  deposit  from  the 
load,  a  patch  of  debris  appearing  on  the  bed  of 
the  trough.  Such  a  patch  travels  slowly  down- 
stream, being  succeeded  after  an  interval  by 
another.  With  suitable  variation  of  conditions 
the  patches  become  more  numerous,  are  regu- 
larly spaced,  and  occupy  a  greater  share  of  the 
bed  surface.  They  may  even  exceed  the  in- 
terspaces in  area.  When  wide  apart  they 
are  usually  shaped  in  gentle  slopes,  but  as  their 
ranks  close  they  assume  the  profiles  of  dunes, 
with  steep  frontal  faces. 

The  advance  of  the  debris  patches  is  like  that, 
of  typical  dunes,  in  that  the  upstream  slopes 
are  eroded  while  the  downstream  slopes  are 
aggraded;  and  in  this  way  then-  travel  is  a 
factor  in  debris  transportation.  It  does  not, 
however,  become  the  dominant  factor,  as  in 


the  dune  phase  of  stream  traction.  There  is 
always  a  large  share  of  the  load  which  passes 
over  the  deposits  without  being  arrested  and 
continues  its  journey  across  the  intervening 
bare  spaces. 

In  this  rhythmic  process  there  is  a  combina- 
tion of  elements  belonging  distinctively  to 
flume  traction  and  to  stream  traction,  and  it  is 
possible  that  the  process  constitutes  a  transition 
from  one  system  to  the  other;  but,  so  far  as  de- 
veloped by  the  experiments,  it  appears  as  a 
phase,  a  rhythmic  phase,  of  flume  traction.  The 
slopes  with  which  it  was  associated  were  much 
flatter  than  those  necessary  to  carry  the  same 
load  by  the  method  of  stream  traction. 

TABLE    OF    OBSERVATIONS. 

In  the  following  table  the  data  are  arranged 
primarily  by  character  of  bed  surface  and  sub- 
ordinately  by  width  of  channel,  discharge, 
grade  of  debris,  and  slope  of  channel  bed. 
The  characters  of  bed  surface  are  illustrated  in 
Plate  III.  Load  is  given  in  grams  per  second, 
width  in  feet,  discharge  in  cubic  feet  per  second, 
and  slope  in  percentage. 


TABLE  67. — Observations  on  flume  traction,  showing  the  relation  of  load  to  slope  and  other  conditions. 
a.  Over  a  surface  of  wood,  planed  and  painted. 


w 

Q 

S 

Value  of  L  for  grade  — 

(C) 

(E) 

(G) 

(II) 

(I) 

(J) 

(EiGO 

(EjHJj) 

(E,H2l3J,.) 

1.00 
1.91 

0.363 

.734 

.303 
.736 

0.32 
.50 
.  75 
1.00 
1.28 
1.50 
1.93 
2.00 
2.50 
2.77 
3.00 
3.41 
3.93 
4.01 
4.05 
4.50 
.32 
.50 
.75 
1.00 
1.28 
1.50 
1.93 
2.00 
2.50 
2.77 
3.00 
3  41 
3.93 
4.01 
4.05 
4.50 
.75 
1.28 
1.93 
2.77 
3.44 
.75 
1.28 
1.93 
2.77 
3.44 

9.1 

6.1 

6.1 

28 

58 

50 

55 

95 

138 

117 

154 

107 

126 

248 

166 

294 
294  ' 

183 
202 

199 

243 
268 
453 
384 
438 
524 

334 

271 
342 

397 

406 

318 
366 
444 

360 

386 
479 

566 

636 

625 

484 

783 

793 

1,092 

1  025 

850 

1,095 

622 

660 
20.5 

685 
18.1 

801 

22.8 

68 

120 

98 

110 

179 

315 

335 

261 

212 

219 

297 

372 

381 

348 

449 
474 
899 
721 
814 
976 

561 

660 
1,248 

778 
570 

427 

909 

949 

590 
640 
771 

580 
726 
765 

1,075 

1,170 

1,595 

790 

1,557 

1,730 

2,360 

1  980 

2,480 

1,278 

1,081 
41.6 
103 
193 
324 
439 
114 
222 
368 
641 
857 

1,127 
58 
117 
222 
392 
471 
147 
248 
430 
725 
846 

1,468 
30 
137 
255 
463 
569 
115 
325 
501 
799 
1,028 

49 
110 
230 

134 
282 
432 

FLUME   TRACTION.  203 

TABLE  67. — Observations  on  flume  traction,  showing  the  relation  of  load  to  slope  and  other  conditions — Continued, 
b.  Over  a  surface  of  wood,  rough-sawn,  unplaced,  and  unpatnted. 


w 

Q 

S 

Value  of  L  for  grade  — 

(E) 

CO) 

(H) 

(I) 

(J) 

(EsHiI,) 

1.00 

0.363 
.734 

1.07 
1.88 
2.11 
2.20 
2.88 
3.09 
3.81 
4.15 
1.07 
1.88 
2.11 
2.20 
2.88 
3.09 
3.81 
4.15 

186 

133 

141 

178 

355 

304 

247 

284 

342 

515 

609 

358 

390 

526 

813 

930 

881 
302 

269 

313 

343 

646 

632 

461 

513 

545 

982 

1,558 

1,550 

673 

720 

772 

1,605 

2,140 

1,665 



c.  Over  a  surface  of  rectangular  wood  blocks,  with  grain  vertical. 


Value  of  L  for  grade  — 

(E) 

(H) 

(I) 

(J) 

(EjHiIs) 

(EjHAJj) 

1.00 

0.734 

2.00 

315 

337 

561 

750 

620 

789 

3.00 

580 

ii!H 

962 

1,397 

1,089 

l,3fiO 

4.00 

911 

1,026 

1,497 

2,060 

1,660 

2,020 

d.  Over  a  pavement  of  sand  grains,  grade  (G),  set  in  cement,  the  debris  being  also  of  grade  (G). 


w 

Q 

s 

L 

1.00 

0.363 

1.00 

12.1 

2.00 

71 

3.00 

175 

4.00 

317 

.734 

1.00 

61.2 

2.  SO 

214 

3.00 

413 

4.00 

656 

e.  Over  a  pavement  of  pebbles,  grades  (H)  and  (I),  set  in  cement. 


w 

Q 

S 

Value  of  L  for  grade  — 

(E) 

(Q) 

(H) 

(D 

(J> 

(E,Gi) 

(EjHiIs) 

(EaHiWi) 

1.00 

0.734 

2.00 
3.00 
4.00 

225 
425 
630 

144 
316 
531 

125 
273 
463 

190 
483 
714 

228 
636 
992 

196 

401 

605 
1,059 

471 

ADJUSTMENT    OF   OBSERVATIONS. 

FORMULATION. 

In  flume  traction,  as  in  stream  traction, 
there  is  a  finite  slope — competent  slope — cor- 
responding to  the  zero  capacity.  An  inspec- 
tion of  the  observational  data  by  plotting 
served  to  show  that  they  could  advantage- 
ously be  adjusted  by  means  of  the  formula 
based  on  the  theory  of  competent  slope: 


Forty-two  observational  series  were  found  to 


give  information  as  to  the  value  of  a.  Of  these, 
26  indicated  positive  values,  three  negative, 
and  the  remainder  values  so  small  as  to  be  of 
uncertain  sign.  The  mean  of  the  42  values  is 
+  0.29  per  cent  of  slope.  Eleven  could  be 
compared  directly  with  values  adopted  in  the 
adjustments  of  stream  traction  data,  the  mean 
of  the  eleven  values  being,  for  flume  traction 
on  a  smooth  surface  +0.14  per  cent,  and  for 
stream  traction  +0.28  per  cent.  This  differ- 
ence is  consonant  with  observed  differences  in 
competent  slope  for  the  two  modes  of  traction. 
With  the  aid  of  this  information,  and  with  use 
of  considerations  connected  with  modifications 


204 


TRANSPORTATION    OP    DEBRIS   BY    RUNNING    WATER. 


of  the  mode  of  flume  traction  by  rough  sur- 
faces, but  here  omitted,  a  scheme  of  values  of  a 
for  the  observational  series  of  Table  67  was 
made  out.  The  adjustments  were  then  made, 
by  the  graphic  methods  described  in  Chapter 
II;  and  their  results  appear  in  Table  68.  The 
same  table  records  the  parameters  of  the  adjust- 
ing equations,  and  also  the  probable  errors. 


No  adjusted  values  are  given  for  grade  (I) 
in  the  first  division  of  the  table,  for  the  reason 
that  the  data,  although  apparently  based  on 
good  observations,  are  strongly  discordant. 
The  observations  were  retained  in  the  record 
because  an  aberrant  fact,  if  established,  may 
prove  peculiarly  valuable;  but  in  this  case  the 
interpretation  has  not  been  discovered. 


TABLE  68. — -Values  of  capacity  for  flujne.  traction,  adjusted  in  relation  to  slope  of  channel. 
a.  Traction  over  a  surface  of  wood,  planed  and  painted. 


w 

Q 

S 

Value  of  C  for  grade— 

(C) 

(E) 

(C) 

(H) 

(I)                (J) 

(E,G,) 

(E,H,I,) 

(EtHJM 

1.00 

0.363 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 

28 
93 
186 
300 
435 
590 
760 
945 

25.5 
74 
134 
205 
283 
366 
453 
553 
654 

25 
87 
158 
234 
315 
398 
486 
575 
665 

29 
93 
170 
255 
350 
450 
558 
666 

138 
266 
410 
568 
732 
910 
1,085 

117 
254 
408 
580 
763 
955 
1,155 

254 
350 
451 
558 
666 
780 

625 
860 
1,085 



Probable  error  (per  cent)  

2.9 

3.7 

0.  8    '          1.1 

1.3 

1.8 

1.2 

Parameters  of  adjusting!*'  '  ' 
equations. 

0.04 
1.63 
100 

0.06 
1.40 

80 

0.25 
1.16 
122 

0.35 
1.22 
138 

1.80 
.96 
520 

0.15 
1.30 
114 

0.25 
1.29 
199 

0.40 
1.28 
223 

:::::::::: 

w 

Q 

a 

Value  of  C  for  grade  — 

(C)              (E) 

(G) 

(H) 

en 

(J) 

(Eid) 

(EjH.Ij) 

(E,HJ»T,) 

1.00 

0.734 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 

46 
177 
345 
548 
790 
1,050 
1,360 
1,700 

54 
145 
258 
382 
518 
665  - 
820 
9S5 
1,150 

50 
151 

268 
393 
527 
668 
815 
965 
1,11D 

65 
179 
310 
453 
610 
770 
940 
1,108 

336 

592 
870 
1,160 
1,465 
1,790 
2,120 

335 
635 
970 
1,310 
1,675 
2,060 
2,450 

338 
470 
645 
830 
1,027 
1,235 
1,450 

790 
1,220 
1,630 
2,030 
2,430 

653 
910 
1,170 
1,440 

Probable  error  (per  cent)  

2.1              1.8 

1.7 

0.7 

0.7 

1.7 

2.7 

0.7 

Parameters  of  adjusting!'  •  •  ' 
equations.                    jj~" 

0.  03            0.  04 
1.60            1.35 
185          \    154 

0.16 
1.20 

187 

0.22 

1.29 
221 

0.65 
1.01 
470 

1.20 
.90 
965 

0.10 
1.24 
202 

0.16 
1.20 
415 

0.28 
1.21 
500 

w 

Q 

a 

Value  of  C  for  grade  — 

Q                 S 

Value  of  C  for  grade— 

(C) 

(E) 

(G)          (H) 

(C)          (E)          (G)          (H) 

1.91 

0.363 

1.0 
1.5 
2.0 
2.5 
3.0 
3.5 

77 
145 
224 

69 
130 
200 
281 
370 
464 

85              75             1.91           0.734              1 
160            171                                                    1 
241             275                                                        2 
328            381                                                    2 
415            491     i                                                3 
507            602                                                    3 

0          197           158 
5          324          281 
0           459           415 
5                         560 

185          204 
320           378 
460          550 
607          730 
755          915 
910         1,095 

0         .                  720 

5                         880 

Probable  error  (pe 

0.5 

3.0          1.5           Probable  error  (per  cent) 

1.4 

3.  3          2.  4 

Parameters  of  adjusting  1°,  •  •• 
equations  1  ?"" 

0.07 
1.44 

86 

0.10 

1.42 
80 

0.30         0.55         „__ 
1.16          1.09 
130            182 

meters  of  adjusting!' 

,...      0.05         0.08         0.20        0.35 
»...      1.16         1.30         1.12         1.06 
il..210           176           238           322 

FLUME   TRACTION. 

TABLE  68. —  Values  of  capacity  for  flume  traction,  adjusted  in  relation  to  slope  of  channel — Continued. 

b.  Traction  over  a  surface  of  wood,  rough-sawn. 


205 


w 

Q 

S 

Value  of  C  for  grade— 

(E) 

(0) 

(H) 

(I) 

(J) 

(E,H,I.) 

1.00 

0.363 

1.5 
2.0 
2.5 
3.0 
3.5 
4.0 

97 
145 
200 
259 
331 
385 

120 

195 
279 
366 
460 
560 

229 
341 
460 
582 
710 
840 

160 
220 
285 
355 
427 

249 
374 
500 
630 
755 

895 

Probable  error  (pe 

•  cent)... 

4.6 

Parameters  of  adjusting!"  •  •  • 
equations.                    j^'" 

0.10 
1.34 

62 

0.30 
1.12 
82 

0.40 

1.28 
106 

1.00 
1.01 
248 

0.30 
1.16 

184 

w 

Q 

S 

Value  of  C  for  grade  — 

(E) 

(G) 

(H) 

(I) 

P) 

(E,HiIt) 

1.00 

0.734 

1.5 
2.0 
2.5 
3.0 
3.5 
4.0 

450 
660 
885 
1,115 
1,360 
1,615 

290 
390 
495 
600 
712 

331 
434 
540 
648 
758 

354 
462 
570 
680 

786 

538 
752 
970 
1,100 
1,325 

1,490 
1,800 
2,100 

Probable  error  (pe 

2.3 

Parameters  of  adjusting]''  •  • 
equations. 

0.08 
1.25 
128 

0.20 
1.11 
173 

0.30 
1.02 
207 

0.65 
1.06 
390 

1.20 
.76 
950 

0.20 
1.18 
325 

c.  Traction  over  wood  blocks  with  grain  vertical. 


d.  Traction  over  pavement  of  sand  grains,  grade  (O). 


Value  of  C. 

Q-0.363 

Q-  0.734 

1.00 

1.0 

11 

61 

1.5 

36 

131 

2.0 

72 

215 

2.5 

119 

310 

3.0 

175 

413 

3.5 

240 

525 

4.0 

311 

650 

Parameters  of  ad-  (a  ... 
justing      equa-^n... 
tions.                   \b\  .  . 

0.50 
1.70 
36 

0.30 
1.72 
101 

Probable  error  (per 

1.1 

0.1 

Value  of  C  for  grade  — 

(E) 

(H) 

(I) 

(J> 

(EiH.Ij) 

(E.HO.J,) 

1.00 

0.734 

2.0 

313 

354 

540 

730 

613 

780 

2.5 

442 

507 

770 

1,090 

860 

1,075 

3.0 

583 

673 

1,008 

1,415 

1,110 

1,390 

3.5 

740 

850 

1,245 

1,740 

1,375 

1,700 

4.0 

900 

1,026 

1,500 

2,040 

1,660 

2,010 

Parameters  of  adjusting  I*'  •  • 
equations. 

0.10 

1.48 
123 

0.45 

1.48 
201 

0.75 
1.05 
430 

1.20 
.81 
880 

0.50 
1.16 
385 

0.60 
1.06 
545 

206 


TRANSPORTATION    OF    DEBRIS   BY   SUNNING    WATER. 

TABLE  68. —  Values  of  capacity  for  fume  traction,  adjusted  in  relation  to  slope  of  channel — Continued, 
e.  Traction  over  pavement  of  pebbles,  grades  (H)  and  (I). 


w 

e 

S 

Value  of  C  for  grade  — 

(E) 

(G) 

(H) 

(I) 

(EiGO 

(EsHA) 

(E3H2I3J2) 

1.00 

0.734 

2.0 
2.5 
3.0 

3.5 
4.0 

220 
313 
416 
529 
650 

143 
220 
310 
412 

522 

122 
192 
272 
363 
460 

200 
325 
460 
600 
740 

250 
420 
605 
800 
1,015 

209 
290 
393 

605 
830 
1,060 

Parameters  of  adjusting  j*  •  '  • 
equations.                      (ft 

0  10 
1.50 
84 

0.40 

1.60 
67 

0.55 

1.52 
68 

1.00 

1.65 
65 

1.00 
1.19 
200 

1.00 
1.26 
250 

1.20 
1.28 
280 

PRECISION. 

Probable  errors  were  computed  from  the 
residuals  of  25  series,  the  residuals  being 
measured  on  the  plots  as  percentages.  The 
greatest  probable  error  computed  for  a  series 
of  adjusted  values  of  capacity  is  ±4.6  per  cent, 
and  their  average  is  ±1.3  per  cent.  The  aver- 
age of  the  25  determinations  of  the  probable 
error  of  an  observation  is  ±3.8  per  cent.  The 
residuals  number  139,  and  their  average  value, 
which  also  is  a  measure  of  the  precision  of  the 
observations,  is  ±4.3  per  cent. 

A  comparison  of  these  measures  with  those 
obtained  from  the  data  for  stream  traction 
shows  that  the  flume-traction  data  are  de- 
cidedly the  more  harmonious.  The  average 
residual  is  more  than  twice  as  great  for  stream 
traction  as  for  flume.  Part  of  this  difference 
may  be  due  to  the  fact  that  the  experiments 
with  flume  traction  came  last  and  had  the 
benefit  of  previous  experience,  but  it  is  to  be 
ascribed  chiefly  to  the  fact  that  in  flume  trac- 
tion the  slope  is  constant,  while  in  stream 


traction   it    is    subject    to    rhythmic    fluctua- 
tions. 

DISCUSSION. 

CAPACITY    AND    CHANNEL    BED. 

Data  illustrating  the  influence  of  the  character 
of  the  channel  bed  on  the  quantity  of  debris 
which  a  stream  can  transport  have  been  as- 
sembled in  Table  69.  They  are  taken  chiefly 
from  the  preceding  table,  but  a  few  items  are 
from  Table  72  and  Table  12.  A  single  item, 
marked  as  interpolated,  is  based  on  a  combi- 
nation of  data  from  Tables  68  and  72.  They 
pertain  to  all  the  simple  grades  and  mixed 
grades  with  which  experiments  were  made  in 
flume  traction;  and  stream  traction  is  repre- 
sented, so  far  as  possible,  by  coordinate  data. 
Comparisons  are  made  for  slopes  of  2  and  3  per 
cent  and  discharges  of  0.363  and  0.734  ft.3/sec. 

The  greatest  capacity  is  in  each  case  asso- 
ciated with  the  smoothest  of  the  tested  channel 
beds,  the  surface  being  that  of  a  plank,  planed 
and  painted,  with  the  grain  running  parallel 
to  the  current. 


TABLE  69. — Comparison  of  capacities  for  flume  traction  associated  with  different  characters  of  channel  bed. 

[Width  of  trough,  1  foot.] 


Q 

S 

Character  of  channel  bed. 

Value  of  C. 

Simple  grades. 

Mixtures. 

(B) 

(C) 

(E) 

(G) 

(H) 

(I) 

(J) 

(EiGO 

(E2HiI3) 

(E3H2I3J2) 

0.363 
.363 
.734 

.734 

2.0 
3.0 
2.0 

3.0 

Planed  wood  

388 

300 

205 
145 

234 

160 

72 
45 

393 
285 
175 

254 
195 

364 
249 

410 
341 

Sawn  wood.  .  . 

Sand  pavement  

Debris  

266 

245 

115 

366 
259 

Planod  wood  

451 
366 

600 
500 

732 
£82 

Sawn  wood  

Sand  pavement.  .  .     . 

Debris  

120 

393 

331 

Planed  wood 

548 

382 
290 
313 

470 
354 
354 

711 
533 
540 

790 

453 

870 
660 
613 

970 

Sawn  wood.  .  . 

Woodblock  

733 

780 

Sand  pavement  

215 
143 
145 

668 
540 

Gravel  pavement.  .  . 

220 
222 

665 
495 
583 

122 

200 

250 

Debris  

483 

Planed  wood  

830 
570 
673 

[1,060] 
970 
1,008 

1,630 
1,490 
1,415 

770 

1,465 
1,150 
1,110 

1,675 

Sawn  wood  

Woodblock... 

1,390 

Sand  pavement  

413 

310 

Gravel  pavement 

416 

272 

209 

460 

605 

605 

FLUME   TRACTION. 


207 


Next  in  order  are  two  varieties  of  unplaned 
wooden  surface;  the  first  being  that  of  boards 
or  planks,  paiallel  to  the  current,  retaining  the 
roughness  left  by  the  saw;  the  second  a  pave- 
ment made  by  sawing  planks  of  Oregon  pine 
into  short  equal  blocks  and  setting  them  on 
edge.  Both  these  surfaces,  as  well  as  those 
described  below,  are  illustrated  in  Plate  III. 
These  two  varieties  proved  to  have  approxi- 
mately the  same  properties  in  respect  to  trac- 
tion, and  the  capacities  associated  with  them 
are  23  per  cent  less  than  those  for  planed 
lumber.  The  range  in  ratio  is  not  large  for 
the  different  experiments,  and  the  value  given 
may  be  taken  as  a  constant  representing  the 
difference  in  efficiency  between  new  unplaned 
and  planed  wooden  flumes.  The  difference 
tends,  however,  to  diminish  with  wear,  the 
unplaned  lumber  becoming  smoother  and  the 
planed  rougher. 

The  next  grade  of  roughness  was  given  by 
coarse  sand — d6bris  of  grade  (G) — set  in 
cement,  so  as  to  constitute  a  pavement  re- 
sembling sandpaper.  The  only  material  run 
over  this  was  debris  of  the  same  grade,  the 
special  purpose  being  to  compare  flume  trac- 
tion with  stream  traction — the  condition  of 
fixed  bed  with  that  of  mobile  bed — when  the 
degree  of  roughness  is  the  same.  The  experi- 
ments gave  the  streams  50  per  cent  greater 
capacity  when  sweeping  the  debris  over  the 
fixed  bed  than  when  moving  it  at  the  same 
slope  by  the  method  of  stream  traction. 

The  sand  pavement  gives  capacities  half  as 
great,  on  the  average,  as  the  surface  of  planed 
lumber,  but  the  contrast  is  stronger  for  the 
smaller  discharge  and  lower  slope  and  less 
marked  for  the  larger  discharge  and  steeper 
slope. 

The  roughest  surface  used,  a  pavement  of 
pebbles  prepared  by  setting  in  cement  a  mix- 
ture of  grades  (H)  and  (I),  gave  still  lower 
capacities.  These  range  from  20  to  62  per 
cent  of  the  corresponding  capacities  given  by 
planed  lumber.  The  obstructing  influence  of 
the  rough  bottom  is  most  strongly  manifested 
when  the  material  transported  has  a  coarseness 
corresponding  to  the  texture  of  the  pavement. 
For  finer  material  its  roughness  is  mitigated  by 
the  lodgment  of  debris,  which  has  the  effect  of 
establishing  a  pavement  of  the  finer  material. 

The  word  "debris"  in  the  table  indicates  a 
channel  bed  composed  of  loose  debris,  the 


debris  in  transit,  and  the  associated  process  is 
that  of  stream  traction.  The  available  data 
afford  comparison  only  for  the  four  finer 
grades,  (B),  (C),  (E),  and  (G),  the  grades 
which  would  be  designated  sand.  Each  com- 
parison, with  an  apparent  exception  to  be 
considered  immediately,  shows  stream  trac- 
tion to  be  less  efficient  than  flume  traction. 
When  stream  traction  is  compared  with  flume 
traction  over  a  smooth  surface,  the  observed 
ratio  of  efficiency  ranges  from  19  to  88  per 
cent,  the  smaller  ratios  being  associated  with 
the  coarser  grades  of  debris. 

The  exception  occurs  when  capacity  over  a 
bed  of  debris  is  compared  with  capacity  over 
a  pavement  of  pebbles,  the  two  capacities 
being  found  to  be  the  same.  The  cases  which 
afford  this  comparison  are  for  grades  (E)  and 
(G),  and  these  fine  materials,  by  filling  the 
hollows  of  the  pavement,  create  a  condition  of 
bed  in  which  stream  traction  dominates.  The 
comparison  is  really  between  normal  stream 
traction  and  stream  traction  modified  by  the 
appearance  of  crests  of  fixed  pebbles  in  the 
channel  bed.  In  harmony  with  tliis  interpreta- 
tion is  the  fact  that  capacities  for  stream  trac- 
tion and  for  traction  over  the  gravel  pavement, 
when  compared  severally  with  capacity  for  trac- 
tion over  smooth  wood,  both  show  contrasts 
which  increase  with  coarseness  of  the  load. 

The  important  general  facts  brought  out  by 
the  comparisons  are  (1)  that  with  a  given  dis- 
charge, channel  width,  and  slope,  the  process 
of  flume  traction  is  able  to  transport  more 
debris  than  that  of  stream  traction,  and  (2)  that 
a  stream's  capacity  for  flume  traction  varies 
inversely  with  the  roughness  of  the  flume  bed. 

The  first  of  these  principles  serves  to  explain 
certain  phenomena  of  clogging.  When  there 
is  fed  to  a  flume  a  load  greater  than  its  stream 
is  able  to  transport,  a  portion  is  deposited. 
This  changes  the  character  of  the  bed  in  such 
a  way  as  to  substitute  stream  traction  for 
flume  traction.  Stream  traction,  being  less 
efficient,  can  carry  still  less  load,  and  a  larger 
fraction  is  deposited.  If  the  conditions  re- 
main unchanged  the  bed  is  built  up  until  its 
slope  becomes  that  necessary  to  carry  the  en- 
tire load  by  stream  traction.  Unless  the  trough 
is  deep  or  short,  overflow  results. 

When  clogging  is  initiated  by  a  temporary 
overloading,  the  stream  loses  power  to  carry  its 
normal  fractional  load,  and  deposition  con- 


208 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


tinues  unless  the  load  is  reduced  considerably 
below  the  normal.  In  practical  operations 
the  first  step  toward  the  abatement  of  a  clog 
is  te  stop  all  feeding  of  load  above  the  deposit, 
that  the  stream  may  be  able  to  take  on  load 
and  thereby  reduce  the  deposit. 

CAPACITY    AND    SLOPE. 

The  rate  at  which  capacity  for  flume  traction 
is  increased  by  increase  of  slope  is  contained 
implicitly  in  the  values  of  n  and  a  assembled  in 
Table  68;  for  n  gives  the  rate  of  variation  of 
capacity  with  8  —  a,  and  the  instantaneous  rate 
of  variation  of  capacity  with  slope  is  given  by 

ij  =  -q .     All  the  tabulated  values  of  n  except 

two  are  greater  than  unity;  and  in  each  of  the 
cases  where  n  is  less  than  unity  all  values  of  \ 
computed  for  the  range  of  slopes  covered  by  the 


experiments  are  greater  than  unity.  The  gen- 
eral fact  is  thus  indicated  that,  within  the  prac- 
tical range  of  conditions,  capacity  increases 
with  slope  in  more  than  simple  ratio.  Effi- 
ciency also  increases  as  slope  increases. 

The  sensitiveness  of  capacity  to  changes  in 
slope  varies  with  changes  of  condition;  and  this 
variation  might  be  illustrated,  as  in  treating  of 
stream  traction,  by  the  tabulation  and  discus- 
sion of  values  of  the  index  of  relative  variation, 
*!-  It  will  suffice,  however,  in  this  case  to 
make  comparisons  by  means  of  the  synthetic 
index,  7t.  Table  70  contains  values  of  that  in- 
dex computed  between  the  limits  of  S  =  2.0  and 
8  =  3.5.  They  represent  42  of  the  51  series  of 
values  given  in  Table  68,  the  other  nine  series 
not  having  sufficient  range  for  the  computa- 
tion. The  arithmetical  mean  of  the  42  values 
of  7t  is  1.46,  and  the  range  is  from  1.08  to  2.08. 


TABLE  70. —  Values  of  /,  for  flume  traction,  computed  between  the  limits  S—2.0  and  S=3.5. 


w 

Q 

Character  of  channel  bed. 

Value  of  /i. 

Simple  grades. 

Mixtures. 

(C) 

(E) 

(G) 

(H) 

(I) 

(J) 

(E,Gi) 

(EjHJ,) 

(EiHjIjJ:) 

1.00 
1.00 

1.91 
1.91 

0.363 
.734 

.363 
.734 

Planed  wood  

1.66 

1.  It 
1.43 
.37 
.30 
.54 

1.30 
1.42 
1.30 
1.20 

.41 
.54 
.40 
.17 
.56 

1.40 

.42 
.32 
.29 

1.52 

Sawn  wood  

1.66 

Planed  wood  

1.66 

1.69 

1.30 

1.35 

Sawn  wood  

1.27 
1.49 

Woodblock  

1.55 

.44 

1.39 

Sand  pavement  

1.60 

Gravel  pavement  

.56 
.50 
.34 

1.89 
1.32 
1.22 

.95 
.40 
1.23 

1.96 

2.08 

Planed  wood  

do  

Inspection  of  these  data  shows,  first,  that 
the  values  are  always  greater  for  Q  =  0.363 
than  for  Q  =  0.734.  The  experiments  deal 
with  no  other  discharges,  but  it  is  probably 
true  in  general  (as  in  case  of  stream  traction) 
that  increase  of  discharge  is  accompanied  by 
decrease  of  the  sensitiveness  of  capacity  to 
slope. 

If  the  index  varies  in  a  systematic  way  with 
fineness  of  d6bris,  its  increase  is  connected 
with  decrease  of  fineness,  but  the  finest  debris 
of  the  table,  (C),  carries  large  values  of  the 
index.  The  apparent  conflict  of  evidence  has 
its  parallel  in  the  fuller  data  for  stream  trac- 
tion (see  p.  108  and  fig.  34),  and  it  is  possible 
that  the  sensitiveness  increases  in  two  direc- 
tions from  a  minimum  value.  Its  variation 
might  in  that  case  be  connected  with  the  law 
relating  capacity  to  fineness,  as  brought  out  in 
a  later  section. 


The  relation  of  the  index  to  roughness  of  bed 
does  not  follow  a  simple  law.  Its  values  are 
in  general  least  for  the  bed  of  rough  lumber  and 
progressively  greater  for  planed  lumber,  wood 
blocks,  and  gravel  pavement. 

The  greater  sensitiveness  of  capacity  to  slope 
when  the  channel  bed  is  a  coarse  pavement 
may  be  connected  with  the  fact  that  the  mode 
of  transportation  over  such  a  bed  is  approxi- 
mately stream  traction;  and  this  suggests  that 
in  flume  traction  the  sensitiveness  may  be  less 
than  in  stream  traction.  Direct  comparison 
can  not  be  made  with  use  of  the  values  of  7, 
in  Table  70,  because  the  slopes  used  in  stream 
traction  experiments  have  less  range;  but  spe- 
cial computations  were  made,  so  far  as  the  data 
were  found  to  overlap.  The  results  are  con- 
tained in  Table  71  and  indicate  that  the  sensi- 
tiveness is  greater  for  stream  traction  than  for 
flume  traction  over  a  smooth  bed,  in  case  of 
grades  (E)  and  (G),  but  less  in  case  of  grade  (C). 


FLUME   TRACTION. 


209 


TABLE  71. — Comparison  of  values  of  Itfor  flume  traction 
over  a  bed  of  planed  wood,  with  corresponding  values  jor 
stream  traction. 


Width 
of  trough 
(feet). 

Grade  of 
debris. 

Dis- 
charge 
(ft.a/sec.). 

Limiting 
values  of 
slope  (per 
cent). 

/ifor 
flume 
traction. 

/i  for 

stream    : 
traction. 

1.00 

(C) 

0.363 

0.5-2.0 

1.71 

1.60 

.734 

.5-2.0 

1.79 

1.46  ' 

(E) 

.363 

.5-2.0 

1.50 

1.99 

.734 

.5-2.0 

1.41 

1.-65 

(G) 

.363 

2.  0-3.  0 

2.27 

2.55 

.734 

1.0-2.0 

1.38 

2.42 

Various  qualifications  and  doubts  being 
omitted,  the  preceding  paragraphs  may  be 
generalized  by  saying  that  the  sensitiveness  of 
capacity  to  slope  is  somewhat  less  in  flume 
traction  than  in  stream  traction.  It  varies  in 
both  directions  from  a  mean  value  expressed 
by  the  exponent  1 .5,  being  greater  as  the  slope 
is  less,  as  the  discharge  is  less,  as  the  fineness 


is  less,  and  as  the  channel  bed  is  rougher. 
Efficiency  for  flume  traction  increases  with 
slope. 

CAPACITY    AND    DISCHARGE. 

Special  series  of  experiments  were  made  to 
determine  the  variation  of  capacity  with  dis- 
charge. In  each  series  the  conditions  of  slope, 
width,  and  grade  of  debris  were  kept  constant 
and  the  discharge  was  varied.  The  observa- 
tions are  given  in  Table  72,  and  the  same  table 
contains  the  adjusted  values  of  capacity,  to- 
gether with  the  parameters  of  the  adjusting 
equations.  Inspection  of  logarithmic  plots 
showed  the  propriety  of  adjusting  by  means  of 
the  formula  used  with  the  data  for  stream 
traction, 

<7=&3«2-K)° (64) 

and  the  computations  were  graphic. 


TABLE  72. — Observations  and  adjusted  data  illustrating  the  relation  of  capacity  for  flume  traction  to  discharge ,  for  a  rectangular 

flume  of  planed  wood  1  foot  wide. 

[L,  observed  load;  C,  adjusted  value  of  capacity;  Q,  discharge.] 


(B) 

(E) 

(H) 

(I) 

(EjHiIi) 

(E) 

(H) 

(I) 

(Ejllilj) 

Q 

L 

L 

L 

L 

L 

L 

L 

L 

L 

0  039 

28.0 

25.0 

.093 

54.5 

79 

128 

50.6 

87 

.182 

151 

96 

136 

110 

144 

169 

228 

272 

325 

.545 
.734 

619 

322 
446 

417 
546 

528 
738 

617 
907 

580 
785 

635 
914 

922 
1,170 

718 
1,147 
1,550 

Parameters  of  adjusting  equations  .. 

K 
0 

63 

0.002 
1.34 
1,430 

0.9 

0.007 
1.09 
630 

0.2 

0.040 

.88 
760 

0.4 

0.140 
.68 

1,000 

0.9 

0.000 
1.05 
1,345 

0.5 

0.004 
1.10 
1,125 

0.3 

0.025 

.88 
1,175 

1.0 

0.080 
.79 
1,650 

0.7 

0.-040 
.88 
2.220 

1.4 

Q 

C 

C 

0 

C 

C 

C 

C 

C 

C 

0.039 

28 

27 

093 

58 

79 

108 

53 

123 

.182 

145 

95 

134 

116 

146 

169 

230 

270 

325 

.363 
.545 
.734 

371 
640 

205 
322 
445 

415 
549 

548 
711 

«27 
890 

572 
795 

450 
670 
870 

600 
880 
1  170 

730 
1,130 
1  550 

Several  values  of  capacity  in  Table  72  agree 
as  to  conditions  with  values  in  Table  68,  and 
these  values  would  be  identical  if  the  experi- 
ments were  homogeneous.  A  comparison 
shows  that  the  values  given  by  the  experiments 
comparing  capacity  and  discharge  are  in  gen- 
eral the  greater,  the  average  difference  being 
6  per  cent.  This  is  evidently  of  the  nature  of 
systematic  error  and  is  probably  connected 
with  some  change  in  apparatus  or  in  detail  of 
20921°— No.  8&— 14 14 


experimental  method  which  occurred  between 
the  making  of  the  two  groups  of  experimental 
series. 

The  sensitiveness  of  capacity  to  changes  of 
discharge  varies  with  conditions.  It  is  greater 
as  the  discharge  is  less,  as  the  slope  is  less,  and 
as  the  channel  bed  is  rougher.  It  is  relatively 
great  for  the  coarsest  and  finest  of  the  de'bris 
used  and  less  for  intermediate  grades.  Un- 
der similar  conditions  it  is  less  for  flume  trac- 


210 


TRANSPORTATION    OF    DEBRIS   BY   RUNNING    WATER. 


tion  than  for  stream  traction.  Expressed  as 
an  exponent,  73,  its  average  value  for  the  range 
of  the  experiments  recorded  in  Table  72  is  1 .26. 
Values  of  73  were  also  obtained  from  data  in 
Table  68  by  comparing  the  capacities  for  Q  — 
0.363  ft.3/sec.  with  those  for  <>  =  0.734  ft.3/sec.; 
and  the  mean  of  such  values  is  0.97.  These 
mean  values  are  not  necessarily  inconsistent, 
for  the  synthetic  index  varies  with  the  range  in 
discharge  for  which  it  is  computed  and  is  lower 
as  the  discharges  are  higher;  but  a  study  of  in- 
dividual values  shows  that  under  identical  con- 
ditions the  data  of  Table  68  give  the  lower  esti- 
mates of  sensitiveness.  The  data  as  a  whole 
indicate  that,  for  the  range  of  conditions  real- 
ized in  the  experiments,  the  average  value  of 
the  exponent  expressing  sensitiveness  to  dis- 
charge is  1.2.  There  can  be  no  question  that 
for  the  larger  discharges  used  it  falls  below 
unity. 

As  the  sensitiveness  of  the  duty  of  water,  and 
also  the  efficiency,  to  discharge  is  expressed  by 


an  exponent  which  is  less  by  unity  than  the 
corresponding  exponent  for  the  sensitiveness  of 
capacity  to  discharge,  it  follows  that  duty  and 
efficiency  vary  little  with  discharge.  In  general 
they  gain  slightly  with  increase  of  discharge, 
but  they  lose  when  the  discharge  or  slope  is 
relatively  large.  This  accords  with  a  result 
obtained  by  G.  A.  Overstrom/  who  found  from 
experiments  with  launders  that  duty  rose  with 
increase  of  depth  to  a  limited  extent  only. 

CAPACITY    AND    FINENESS. 

In  Tables  68  and  72  the  values  of  capacity 
standing  in  any  horizontal  line  constitute  a 
series  illustrating  the  variations  which  are  re- 
lated to  grades  of  debris,  and  if  those  in  the 
columns  for  mixtures  be  excepted  they  illus- 
trate the  relations  of  capacity  to  fineness. 
Table  73  contains  a  selection  of  data  from  those 
tables,  together  with  a  single  line  taken  from 
Table  69. 


TABLE  73. — -Values  of  capacity  for  flinne  traction,  illustrating  the  control  of  capacity  by  fineness  of  debris. 


Character  of  channel  bed. 

w 

Q 

S 

Valu»  of  C  for  grade  — 

(B) 

(C) 

(E) 

(G) 

(H) 

(I) 

(J) 

.91 
.00 
.00 
.00 
.00 

.00 
.00 
.00 
.00 

0  734 
363 
363 
734 
734 

734 

734 
734 
363 

2.0 
2.0 
3.0 
2.5 
3.0 

3.0 
3.0 

459 

415 
202 
366 
518 
665 

495 

583 
416 
115 

460 

550 
268 
451 
645 
830 

570 
673 
272 

388 

383 

590 
790 
1,050 

398 
527 
668 

540 

625 
1,220 
1,630 

1,490 
1,415 

653 
910 

970 
1,008 
200 

Wood  block 

310 

45 

De'bris  (stream  tra  ction)  

2.0 

266 

245 

As  the  tables  are  examined,  one  of  the  fea- 
tures arresting  attention  is  that  in  most  of  the 
series  the  smallest  value  of  capacity  does  not 
appear  at  one  end  of  the  line  but  at  some  inter- 
mediate point.  The  occurrence  of  a  minimum 
is  in  fact  characteristic  of  all  tested  varieties 
of  flume  traction  except  that  in  which  the  bed 
is  a  pavement  of  pebbles.  To  give  the  feature 
graphic  expression  the  data  of  the  last  five 
lines  of  Table  73  are  plotted  in  figure  69,  where 
the  horizontal  scale  is  that  of  linear  fineness,  F. 
The  plotted  points  are  far  from  regular,  but 
the  general  character  of  the  representative 
curves  is  unmistakable,  and  freehand  lines  have 
been  drawn.  On  another  sheet,  not  reproduced, 
the  same  data  were  plotted  in  relation  to  mean 
diameter  of  particles — the  reciprocal  of  F— 
with  similar  result,  except  that  the  lower  two 
curves  became  concave  upward. 


These  curves  illustrate  the  most  important 
difference  between  the  laws  of  flume  traction 
and  those  of  stream  traction.  In  stream  trac- 
tion capacity  increases  continuously  as  fineness 
increases.  In  flume  traction  capacity  increases 
with  fineness  when  the  grades  of  debris  com- 
pared are  relatively  fine  but  increases  with 
coarseness  when  the  grades  are  relatively 
coarse.  So  far  as  these  experiments  show,  the 
minimum  of  capacity  corresponds  to  a  coarse 
sand,  but  its  position  on  the  scale  of  fineness 
may  be  assumed  to  vary  with  slope,  discharge, 
and  roughness  of  bed. 

The  curve  for  flume  traction  over  a  bed 
paved  with  gravel  shows  no  minimum  but  is 
of  the  same  type  as  the  curve  for  stream  trac- 
tion. This  fact  is  confirmatory  of  an  inference 

1  Quoted  by  R.  H.  Richards  in  Ore  dressing. 


FLUME    TRACTION. 


211 


already  drawn  (p.  207),  that  the  transportation 
of  fine  debris  over  a  fixed  bed  of  coarser  debris 
particles  is  essentially  of  the  nature  of  stream 
traction.  It  may  fairly  be  inferred  that  if  we 
were  able  to  extend  this  curve  into  the  region 
of  debris  coarser  than  the  gravel  of  the  bed,  a 
minimum  would  be  developed. 

If  the  curves  were  to  be  traced  toward  the 
right,  by  means  of  additional  experiments  with 


1,400 


200 
Linear  fineness 


600 


FIGURE  69. — Curves  illustrating  the  relation  of  capacity  for  flume  trac- 
tion to  fineness  of  de'bris.  Data  from  bed  of  planed  wood  are  recorded 
by  crosses;  from  wood-block  pavement  by  circles;  other  data  by  dots. 

finer  de'bris,  there  can  be  little  doubt  that  they 
would  be  found  to  continue  their  ascent ;  but 
eventually,  as  curves  of  traction,  they  would 
come  to  an  end  with  the  passage  of  the  process 
of  transportation  from  traction  into  suspension. 
In  the  opposite  direction  they  may  be  conceived 
to  attain  a  maximum  and  then  drop  suddenly 
to  the  base  line;  for  despite  the  law  of  increase 
of  capacity  with  coarseness,  there  must  be  a 
degree  of  coarseness  for  winch  the  force  of  the 
current  is  not  competent,  and  when  that  is 
reached  the  ordinate  of  capacity  becomes  zero. 
The  position  of  this  limit,  which  I  have  in  earlier 
pages  called  competent  fineness,  evidently 
depends  on  slope  and  discharge,  as  determining 
the  force  of  the  current,  and  on  the  degree  of 
rounding  of  the  de'bris. 

The  double  ascent  of  the  curve  of  flume 
traction  is  susceptible  of  plausible  explanation, 
by  means  of  considerations  connected  with  the 


process  of  rolling.  The  process  of  rolling  in- 
volves a  question  of  space.  Each  rolling  pebble 
occupies  an  area  of  the  channel  bed  somewhat 
larger  than  its  sectional  area,  even  if  the 
pebbles  are  arranged  in  the  closest  possible 
order.  If  we  conceive  the  channel  bed  to  be 
occupied  by  rolling  pebbles  of  a  particular  size, 
separated  by  spaces  which  bear  a  definite  ratio 
to  the  diameters  of  the  pebbles;  and  again 
conceive  it  to  be  occupied  by  rolling  pebbles 
of  a  larger  size,  with  the  same  ratio  between 
interspace  and  diameter;  it  is  evident  that  the 
total  volumes  or  masses  of  pebbles  in  the  two 
cases  will  be  proportional  to  the  diameters. 
If  the  larger  pebbles  have  twice  the  diameter 
of  the  smaller,  then  a  given  area  of  bed  will 
contain  twice  as  much  rolling  load  of  the  larger 
pebbles  as  of  the  smaller.  It  is  also  true,  as 
stated  on  an  earlier  page,  that  the  rolling  speed 
is  somewhat  greater  for  larger  pebbles  than  for 
smaller.  The  tendency  of  these  two  factors  is 
the  same,  to  make  the  load  greater  for  large 
particles  than  for  small,  when  the  process  of 
transportation  is  rolling.  The  analysis  is  doubt- 
less too  simple — the  degree  of  crowding  on  the 
bed,  for  example,  may  not  be  the  same  for 
different  sizes,  and  the  degree  of  crowding  may 
affect  the  speed  of  rolling— but  qualifying 
factors  can  hardly  impair  the  qualitative 
inference  that  the  rolling  load  increases  with 
coarseness. 

It  is  a  matter  of  observation  that,  under 
similar  conditions  determining  force  of  current, 
the  dominant  process  in  flume  traction  is  for 
coarse  debris  rolling  and  for  fine  debris  salta- 
tion. When  the  process  is  rolling,  as  just 
shown,  capacity  increases  with  coarseness  of 
de'bris.  When  it  is  saltation,  as  illustrated  by 
the  body  of  experiments  on  stream  traction, 
capacity  increases  with  fineness.  With  the 
passage  from  saltation  to  suspension  the  effect 
is  even  heightened,  and  it  is  probable  that  in 
a  number  of  the  recorded  experiments  the 
process  was  largely  that  of  suspension.  Thus 
the  double  ascent  of  the  capacity-fineness  curve 
is  determined  by  the  distinctive  properties  of 
two  (or  three)  modes  of  propulsion. 

If  the  preceding  explanation  is  well  founded, 
the  nature  of  the  law  connecting  capacity  with 
the  degree  of  comminution  of  the  debris  in  any 
particular  case  depends  on  those  conditions 
which  determine  the  dominant  process  of  con- 
veyance. If  the  channel  bed  is  smooth,  and  if 


212 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


slope  and  discharge  are  so  adjusted  as  to  give 
a  moderate  velocity,  the  progression  of  sand 
may  be  by  rolling,  and  in  that  case  the  capacity 
for  different  sands  will  vary  inversely  with  their 
fineness.  But  if  over  the  same  smooth  bed  the 
current  runs  swiftly,  sand  will  be  made  to 
travel  by  saltation,  or  by  saltation  and  suspen- 
sion, and  then  the  capacity  for  different  sands 
will  vary  directly  with  their  fineness.  The  par- 
ticular velocity  with  which  the  function  re- 
verses will  depend  on  the  quality  of  the  bed, 


being  lower  if  the  bed  is  somewhat  rough, 
because  roughness  changes  rolling  to  saltation. 
The  critical  velocity  will  be  higher  for  gravel 
than  for  sand,  because  higher  velocity  is 
needed  to  make  coarser  debris  leap. 

The  experiments  which  have  been  made  were 
not  sufficiently  varied  to  afford  test  for  these 
inferences,  and  as  there  is  no  present  oppor- 
tunity for  continuance  of  laboratory  work  the 
inferences  must  be  regarded  as  largely  hypo- 
thetic. 


TABLE  74. — Capacities  for  flume  traction  of  mixed  grades  and  their  component  simple  grades. 


Character  of  bed. 

Q 

Mixed^grade. 

Capacity 
for 
mixture 
(gm./sec.). 

Capacity  for  single  grade  (gm./sec.). 

(E) 

(G) 

(H) 

(I) 

(J) 

0.363 

(F,GO 

450 
770 
1,465 
1,675 
582 
1,115 
1,110 
1,390 
460 
605 
605 

225 

366 

385 
665 

586 
665 

502 
665 

233 
259 

446 
495 

444 
583 

417 
583 

230 
416 

242 
416 

181 
416 

225 
398 

385 
668 

Do 

.734 
.734 

(EiGi)  

Do 

(EsHiIj)  

293 
830 

335 
830 

116 
366 

223 
570 

222 
673 

278 
673 

586 
910 

502 
910 

233 
500 

446 
970 

444 
1,008 

417 
1,008 

Do                                                           

.734 

335 
1,630 

.363 

(EsHiIs).   .. 



Do  

Wood  block 

.734 
.734 
.734 
.734 
734 

(EjHiIj)  
(EjHiIz)  

Do                      

(E,H,I|Jt)  
(EiGi)  

230 
310 

278 
1,415 

Do 

121 
272 

121 
272 

242 

209 

181 
209 

Do 

.734 

(E,H2ISJ!)  

121 

MIXTURES. 

Three  mixtures  of  simple  grades  of  debris 
were  treated  in  the  laboratory.  The  compo- 
nents of  one,  (E^),  were  a  medium  sand  and 
a  coarse  sand.  The  others,  (E2HjI2)  and 
(E3H2I3J2),  combined  medium  sand  with  fine 
gravel  and  coarser  gravel.  All  three  were 
tested  in  relation  to  slope  on  the  smooth  chan- 
nel bed  and  on  the  gravel  pavement — one  on 
the  rough-sawn  bed  and  two  on  the  bed  of 
wooden  blocks.  One  entered  into  the  experi- 
ments on  capacity  in  relation  to  discharge. 
The  results  are  contained  in  Tables  67,  68,  and 
72.  The  data  from  those  tables  which  pertain 
to  a  channel  width  of  1  foot  and  a  channel  slope 
of  3  per  cent  are  assembled  in  Table  74.  The 


table  contains  also  the  capacity  quota  for  each 
constituent,  computed  from  the  capacity  for 
the  mixture;  and  beneath  each  of  these  quotas 
is  printed  the  capacity  for  the  constituent  when 
the  entire  load  is  composed  of  it. 

A  general  fact  brought  out  by  this  table  is 
that  the  current  can  transport  more  of  a 
mixture  than  it  can  of  any  one  of  the  constit- 
uent grades.  The  table  records  a  single  excep- 
tion, the  capacity  for  the  mixture  being  1,390 
gm./sec.,  while  that  for  its  coarsest  constituent 
grade  is  1,415  grams. 

Another  general  fact  shown  is  that  the  ca- 
pacity for  each  component  as  part  of  a  mixture 
is  less  than  the  capacity  for  the  same  compo- 
nent if  transported  separately.  To  this  also 
there  is  a  single  apparent  exception,  but  as  it 


FLUME   TRACTION. 


213 


occurs  among  the  data  for  traction  over  a  gravel 
pavement  it  illustrates  a  feature  of  stream  trac- 
tion rather  than  flume  traction. 

The  two  principles  may  be  illustrated  to- 
gether by  saying  that  if  a  stream  is  carrying  its 
full  load  of  a  grade  narrowly  limited  in  range 
of  fineness,  and  a  different  grade  of  debris  is 
added,  the  total  load  is  thereby  increased,  but 
this  increase  is  accompanied  by  a  diminution 
of  the  quantity  carried  of  the  first-mentioned 
grade.  In  contrast  with  this  is  the  law  found 
for  stream  traction — that  the  load  of  the  ini- 
tially transported  grade  is  increased  by  the 
moderate  addition  of  other  debris,  provided 
the  added  d6bris  is  relatively  fine. 

The  difference  between  the  two  cases  is 
thought  to  be  connected  with  rolling.  In 
flume  traction  over  a  smooth  bed  the  path  for 
rolling  particles  is  roughened  by  the  presence 
of  smaller  particles.  In  stream  traction  the 
pathway  for  larger  particles  is  smoothed  by  the 
presence  of  smaller  particles  and  rolling  is 
promoted. 

In  stream  traction  the  capacity  for  a  mixture 
is  determined  chiefly  by  the  capacity  for  its 
finer  components,  and  as  mean  fineness  also 
depends  chiefly  on  the  fineness  of  the  finer 
componejits,  mean  fineness  is  a  serviceable 
gage  of  capacity.  In  flume  traction  the  rela- 
tion is  quite  different.  Because  of  the  double 
ascent  of  the  curve  of  capacity  and  fineness, 
it  may  readily  occur  that  the  capacity  for  a 
mixture  is  most  nearly  related  to  that  for  the 
coarsest  component — in  fact,  that  is  true  of 
the  three  mixtures  tested  in  our  experiments — 
and  when  that  is  the  case  there  is  no  parallelism 
between  capacity  and  mean  fineness. 


CAPACITY    AND    FORM   RATIO. 

The  data  bearing  on  the  relations  of  capacity 
to  the  depth  and  width  of  current,  and  their 
ratio,  are  meager.  Most  of  the  experiments 
were  conducted  with  a  single  trough  width,  1 
foot.  The  only  other  width  used  was  1.91 
feet,  and  its  use  was  associated  with  but  four 
grades  of  debris  and  a  single  character  of 
channel  bed — the  smoothest. 

Depths  of  current  were  not  in  general  meas- 
ured during  the  passage  of  loads,  because  the 
surfaces  of  load-bearing  currents  were  usually 
so  rough  as  to  make  good  determinations  im- 
possible. Good  measurements  were  made  of 
unloaded  streams,  and  the  results  are  here 
tabulated.  Attempts  to  measure  depths  of 
loaded  streams  yielded  one  result  thought 
worthy  of  record.  With  a  discharge  of  0.734 
ft.3/sec.,  a  width  of  1  foot,  a  slope  of  4  per  cent, 
and  a  full  load  of  debris  of  grade  (E3IT2I3J2), 
the  depth  was  9  per  cent  greater  than  for  the 
corresponding  unloaded  stream. 

Table  76  compares  the  capacities  found  for  a 
trough  width  of  1.91  feet  with  corresponding 
capacities  for  a  width  of  1  foot.  By  aid  of 
Table  75  it  brings  capacities  into  relation  also 
with  depths. 

TABLE  75. — Depths  and  form  ratios  of  unloaded  streams,  in 
troughs  of  wood,  planed  and  painted. 


to—  1.91 

w-1.00 

Q 

S 

d 

R 

d 

R 

0.363 

1 

0.076 

0.040 

0.120 

0.120 

2 

.062 

.032 

.096 

.096 

3 

.050 

.029 

.082 

.082 

.734 

1 

.119 

.002 

.194 

.194 

2 

.098 

.051 

.154 

.154 

3 

.086 

.045 

.130 

.136 

TABLE  76.— Capacities  for  flume  traction  in  troughs  of  different  wullhs. 


Q 

S 

Grade  (C). 

Grade  (E). 

Grade  (G). 

Grade  (H). 

w—  1.91      w-1.00 

w-1.91 

to-  1.00 

tt=1.91 

1C-  1.00 

to-  1.91 

W-1.00 

Values  of  C. 

0.363 

.734 

.363 
.734 

1.0 
2.0 
3.0 
1.0 
2.0 
3.0 

1.0 
2.0 
3.0 
1.0 
2.0 
3.0 

n          93 

224              300 

69 
200 
370 
158 
415 
720 

74 
205 
366 
145 
382 
665 

85 
241 
415 
185 

460 
755 

87 
234 
398 
151 
393 
668 

275 
491 

254 
451 

197              177 
459              543 

550 
915 

470 
830 

C...I 

ClM 

0.83 

.75 

0.93 
.98 
1.01 
1.09 
1.09 
1.08 

0.98 
1.03 
1.04 
1.22 
1.17 
1.13 

1.08 
1.08 

1.11 

.89 

1.17 
1.10 

214 


TRAKSPORTATION    OF    DEBRIS   BY   RUNNING    WATER. 


Simple  combinations  of  the  quantities,  into 
which  it  is  not  necessary  to  enter,  show  (1) 
that  with  constant  depth,  capacity  increases 
with  width  and  more  rapidly  than  width,  and 
(2)  that  with  constant  width,  capacity  increases 
with  depth  and  more  rapidly  than  depth.  The 
rate  of  increase  with  depth,  if  expressed  as  an 
exponent,  may  be  as  low  as  1.2  or  as  high  as 
2.0.  From  these  premises,  a  line  of  reasoning 
parallel  to  that  of  Chapter  IV  shows  that  for 
flume  traction,  as  for  stream  traction,  the 
function  C=f(R)  increases  to  a  maximum 
and  then  decreases.  The  value  of  R  corre- 
sponding to  maximum  capacity — the  optimum 
form  ratio — can  not  be  determined  from  obser- 
vations involving  but  two  channel  widths,  but 
its  limits  can  in  some  cases  be  indicated.  For 
example,  when  the  capacities  conditioned  by 
the  same  slope  and  discharge  are  approximately 
the  same  for  the  two  trough  widths,  it  may  be 
inferred  that  the  optimum  falls  between  the 
values  of  R  associated  with  the  two  capacities. 
Thus  for  grades  (E),  coarse  sand,  with  a  slope 
of  2  per  cent,  the  capacities  are  about  the 
same  for  the  two  widths,  while  the  form  ratios 
are  0.032  and  0.096,  or  one-thirtieth  and  one- 
tenth;  and  it  is  inferred  that  the  optimum  form 
ratio  falls  between  those  fractions. 

From  inferences  of  this  sort,  used  in  combi- 
nation with  the  general  principles  of  flume 
traction,  a  number  of  tentative  conclusions 
have  been  drawn.  They  are  of  so  hypothetic 
a  character  that  the  reasoning  connected  with 
them  is  not  thought  worthy  of  record,  but  the 
conclusions  themselves  may  perhaps  be  of 
some  service,  until  replaced  by  others  of  more 
secure  foundation.  They  are:  The  ratio  of 
depth  to  width  giving  a  current  the  highest 
efficiency  for  flume  traction  (1)  is  greater  for 
gentle  slopes  than  for  steep,  (2)  is  greater  for 
small  discharge  than  for  large,  (3)  is  greater  for 
fine  debris  than  for  coarse,  (4)  is  greater  for 
rough  than  for  smooth  channel  beds,  and  (5) 
is  in  general  less  than  for  stream  traction. 
The  first  and  second  propositions  apply  also  to 
stream  traction.  The  third  is  the  reverse  of 
the  relation  determined  for  stream  traction, 
and  its  applicability  may  be  limited  to  condi- 
tions under  which  rolling  is  the  dominant  mode 
of  transit.  By  aid  of  the  fifth  and  fourth 
Table  31  may  be  roughly  applied  to  practical 
problems  of  flume  traction. 


TROUGH    OF    SEMICIRCULAR    CROSS    SECTION. 

A  few  experiments  were  made  with  a  trough 
of  galvanized  sheet  iron,  1  foot  wide,  having  a 
semicircular  section.  It  was  given  a  slope  of 
1  per  cent,  and  in  it  were  tested  four  grades  of 
debris.  The  observational  data  appear  in 
Table  77.  The  object  of  the  experiments  was 
to  determine  whether  a  channel  with  curved 
perimeter  is  more  efficient  or  less  efficient 
than  one  with  rectangular  section;  the  capaci- 
ties obtained  are  compared,  in  the  table,  with 
those  determined  for  a  flat-bottomed  trough 
of  the  same  width.  As  the  discharge  used  did 
not  fill  the  semicylindric  trough,  the  width  of 
water  surface  was  less  than  1  foot.  The 
width  of  channel  bed  occupied  by  the  load 
ranged  from  one-fifth  to  one-half  of  the  width 
of  water  surface.  The  medial  depth  of  water 
was  greater  than  in  the  rectangular  channel, 
and  the  mean  velocity  was  higher.  The  higher 
velocity  is  a  factor  favorable  to  the  develop- 
ment of  capacity;  the  narrower  field  of  traction 
is  an  unfavorable  factor.  The  resultant  of  the 
two  was  unfavorable,  the  capacities  for  the 
semicylindric  trough  being  only  half  as  great 
as  for  the  rectangular.  The  result  is  qualified 
by  the  fact  that  the  troughs  compared  were 
not  of  the  same  material,  but  the  disparity  of 
capacities  is  too  great  to  be  ascribed  to  that 
factor. 

TABLE  77. — Data  on  flume  traction  in  a  semicylindric  iron 
trough  of  0.5  foot  radius:  with  comparative  data  for  a  rec- 
tangular wooden  trough  1  foot  wide. 


Slope 
(per  cent). 

Discharge 
(ft.«/sec.). 

Grade  of 
debris. 

Capacity  (gm./sec.). 

In  semi- 
cylindric 
trough. 

In  rec- 
tangular 
trough. 

1.00 

0.363 

.734 

(C) 

(F 

(]•: 

(F 
(G) 

43 
88 

37 
64 
66 

74 

93 

74 

145 

151 

Depth      ( 
stream  ( 
Width  of 
unloade< 
Width  of 
(foot) 

if      unloaded  IQ-  0.363... 
foot).              \Q=0.73» 

0.181 
.257 
.770 
.874 

.17  to.  37 
3.85 
4.60 

0.120 
.194 
1.00 
1.00 

1.00 

3.02 
3.78 

water  surfaceJQ™  0.363... 
lstream(foot)\Q=  0.734... 
be     occupied  by    oad 

Mean    velocity    of   un-/Q-0.3R3... 
loaded  stream(ft./sec.)\Q=0.734... 

The  fact  that  the  doubling  of  the  discharge 
did  not  double  the  observed  load  indicates  that 
the  duty  of  water  diminishes  as  discharge 


FLUME   TRACTION. 


215 


increases,  but  the  data  are  too  few  to  give 
confidence  to  the  inference.  The  three  capaci- 
ties with  a  discharge  of  0.363  ft.3/sec.  indicate 
a  minimum  in  the  curve  of  C=f(F),  thus  sup- 
porting the  generalizations  already  made  from 
the  results  with  rectangular  troughs. 

The  low  efficiency  here  found  for  an  open 
channel  having  a  circular  arc  for  its  perimeter 
suggests  that  the  cylindric  form  commonly 
given  to  closed  conduits  for  the  hydraulic  con- 
veyance of  debris  may  not  be  the  most  efficient. 

A  second  suggestion  is  connected  with  the 
fact  that  the  semicylindric  trough,  while  it  nar- 
rows the  field  of  traction,  at  the  same  time  gives 
a  high  velocity  to  the  water.  It  thus  concen- 
trates the  available  force  and  energy  on  the 
narrow  field.  Though  the  result  is  not  favor- 
able to  capacity,  it  may  be  favorable  to  com- 
petence. When  but  a  small  load  is  to  be  trans- . 
ported,  the  practical  problem  may  be  one  of 
competent  velocity ;  and  such  a  trough  appears 
well  adapted  to  the  production  of  competent 
velocity  with  economy  of  discharge  and  slope. 

SUMMARY. 

Iii  the  transportation  of  d6bris  in  flumes 
much  of  the  movement  is  usually  by  rolling 
and  sliding.  This  is  especially  true  if  the  cur- 
rent is  gentle  or  the  debris  coarse.  With  a 
very  swift  current  or  with  fine  debris  the  par- 
ticles travel  by  a  series  of  leaps,  and  with  the 
finest  debris  the  load  is  suspended.  When  the 
conditions  are  such  that  the  principal  move- 
ment is  by  rolling  and  sliding,  the  capacity  of 
the  current  increases  with  the  coarseness  of  the 
debris  transported,  this  law  holding  good  up 
to  the  limit  of  coarseness  at  which  the  current 
is  barely  competent  to  start  the  particles. 
When  the  conditions  are  such  that  the  princi- 
pal movement  is  by  saltation,  the  capacity  of 
the  current  increases  with  the  fineness  of  the 
debris,  the  law  holding  good  up  to  and  prob- 
ably beyond  the  critical  fineness  at  which  the 
current  is  competent  to  carry  the  debris  in 
suspension. 

Under  all  conditions  the  capacity  is  increased 
by  steepening  the  slope,  and  the  increase  of  ca- 
pacity is  more  rapid  than  the  increase  of  slope. 
The  capacity  may  vary  with  a  power  of  the 
slope  as  low  as  the  1 .2  power,  or  with  one  higher 
than  the  second.  A  general  average  for  the 
experimental  determinations  is  the  1.5  power. 


Under  all  conditions  the  capacity  is  increased 
by  enlarging  the  discharge.  It  may  be  in- 
creased in  the  same  ratio,  in  a  higher  ratio,  or 
in  a  somewhat  lower  ratio. 

The  highest  capacity  is  associated  with  the 
smoothest  channel  bed.  Progressive  increase 
of  roughness  reduces  capacity  progressively  un- 
til the  texture  of  the  bed  becomes  coarser  than 
the  debris  of  the  load.  The  mode  of  transpor- 
tation then  passes  from  flume  traction  to  stream 
traction.  Under  like  conditions  of  slope,  dis- 
charge, and  character  of  debris,  flume  traction 
gives  higher  capacities  than  stream  traction. 

Rectangular  or  box  flumes  have  higher  ca- 
pacity than  semicylindric  flumes  of  similar 
width.  Up  to  a  limit,  which  varies  with  con- 
ditions, the  capacity  is  enlarged  by  increasing 
the  width  of  channel  at  the  expense  of  depth  of 
current.  The  ratio  of  depth  to  width  which 
gives  highest  efficiency  has  not  been  well  cov- 
ered by  the  experiments,  but  it  is  believed  to 
be  rarely  greater  than  1:10  and  often  as  small 
as  1 : 30.  For  large  operations  the  determina- 
tion of  width  will  usually  represent  a  compro- 
mise between  efficiency  and  the  cost  of  con- 
struction and  maintenance. 

As  most  of  the  experiments  were  made  with 
sorted  debris,  each  grade  being  nanowly 
limited  as  to  range  in  the  size  of  its  particles, 
and  as  most  practical  work  is  with  aggregations 
having  great  range  in  size,  the  loads  and 
capacities  here  reported  need  qualification.  By 
experiments  with  mixtures  of  the  laboratory 
grades  it  was  found  that  the  load  carried  of  a 
mixture  is  greater  than  the  load  of  any  one  of 
its  important  components  taken  separately. 
It  is  in  general  true  that  the  capacities  for  com- 
plex natural  grades  of  debris  are  greater  than 
the  tabulated  capacities  for  the  laboratory 
grades  they  most  nearly  resemble. 

COMPETENCE. 

The  experiments  in  flume  traction  were  prac- 
tically limited  in  their  range  by  phenomena  of 
competence,  and  these  limitations  were  of  use 
in  determining  values  of  a,  K,  and  <f>,  but  no 
effort  was  made  to  observe  competence  di- 
rectly and  precisely.  There  are,  however,  a 
few  observations  by  others,  which  may  properly 
be  assembled  here,  although  it  is  not  practicable 
to  use  them  as  checks  on  our  work.  Our  in- 
definite data  pertain  to  slope  and  discharge, 


216 


TRANSPORTATION    OF   DEBRIS   BY   RUNNING    WATER. 


while   the   observations   of  others   pertain   to 
velocity. 

The  experiments  of  Dubuat  *  (1783)  have 
been  assumed,  both  by  him  and  by  others,  to 
pertain  to  stream  traction,  but  his  account  of 
apparatus  and  methods  makes  it  probable  that 
what  he  really  investigated  was  chiefly  com- 
petence for  flume  traction.  He  used  a  trough 
of  plank,  with  the  grain  lengthwise,  and  meas- 
ured the  velocity  of  the  current  by  observing 
the  speed  of  balls  slightly  heavier  than  water 
as  they  were  swept  along  the  bottom.  In  a 
current  of  a  particular  velocity  he  placed 
successively  various  kinds  of  debris  and  noted 
their  behavior,  then  changed  the  velocity  and 
repeated.  His  results  are  as  follows: 

Competent  bed  velocity 

(ft./sec.). 
Potter's  clay Between  0.27  and  035 


0.7 


1.1 


0.35 
0.62 
1.1 

2.1 
3.2 


0.53 

0.7 

1.55 

3.2 
4.0 


Coarse  angular  sand 
River  gravel: 

Size  of  anise  seed 

Size  of  peas 

Size  of  common  beans 

Rounded  pebbles,  1  inch  in  diame- 
ter  

Angular  flints,  size  of  hen's  eggs. . . 

J.  W.  Bazalgette,  in  discussing  the  flushing 
of  sewers  and  therefore  presumably  consider- 
ing flume  traction  rather  than  stream  traction, 
quotes  the  following  results  of  experiments  by 
Robison : 2 

Competent  bed  velocity 
(ft./sec.). 

Fine  sand 0.  5 

Sand  coarse  as  linseed 67 

Fine  gravel 1.0 

Round  pebbles,  1  inch  in  diameter 2.  0 

Angular  stones,  size  of  eggs 3.  0 

In  1857  T.  E.  Blackwell3  conducted  elabo- 
rate experiments  to  determine  the  vel  >cities 
necessary  to  move  various  materials  in  sewers. 
His  channel  was  of  rough-sawn  plank,  60  feet 
long  and  4  feet  wide,  with  level  bottom.  Ve- 
locities were  measured  by  a  tachometer,  but  the 
relation  of  the  velocity  measurements  to  the 
bed  is  not  stated.  The  tests  were  applied  to 
natural  d6bris  of  various  kinds  and  also  to  types 
of  artificial  objects  likely  to  enter  sewers.  The 
objects  were  treated  singly  and  in  aggregates, 
with  the  general  result  that  an  aggregation  re- 

1  Dubuat-Naneay,  L.  G.,  Principes  d'hydraulique,  vol.  1,  p.  100;  vol. 
2,  pp.  57,  79,  95,  Paris,  1786. 

2  Inst.  Civil  Eng.  Proc.,  vol.  24,  pp.  289-290,  1865. 

3  Accounts  and  papers  [London],  Sess.  2, 1857;  Metropolitan  drainage, 
vol.  36,  Appendix  IV,  pp.  167-170,  Pis.  1-5. 


quires  higher  velocity  to  move  it  than  does  a 
single  object.  It  is  evident  that  the  experi- 
ments on  single  objects  pertain  to  flume  trac- 
tion and  some  of  those  on  aggregations  to 
stream  traction.  From  his  tabulated  results 
the  subjoined  data  are  selected  as  representing 
or  illustrating  the  velocities  competent  for 
natural  debris,  the  column  of  mean  diameter 
being  added  by  me.  He  infers  from  the  ex- 
periments that  (1)  for  objects  of  the  same 
character  competent  velocity  increases  with  the 
mass;  (2)  for  objects  of  the  same  size  and  form 
it  increases  with  the  specific  gravity;  (3)  for 
objects  of  different  form  it  is  greater  in  propor- 
tion as  they  depart  from  the  form  of  a  sphere; 
and  (4)  for  objects  in  motion  the  rate  of  travel 
increases  with  the  velocity  of  the  current. 

TABLE  78. — Observations  by  Blackwell  on  velocity  competent 
for  traction. 


Material. 

Volume. 

Mean  di- 
ameter. 

Competent 
velocity. 

Single  objects  (illustrating  flume  trac- 
tion) : 
Brickbat  (roughly  cuboid).. 

Cubic 
inches. 
18.5 

Feet. 
0.27 

Ft./sec. 
1  75-3  00 

Do 

12.98 

24 

2  25-2  50 

Do... 

13.6 

.25 

2  00-2  25 

Do 

7  33 

20 

2  00-2  25 

Do... 

4.76 

.17 

2  25-2  50 

Do 

2  59 

14 

1  75-2  00 

10.37 

.22 

3  00-3  25 

Do 

6  05 

19 

2  00-2  25 

Do 

4.11 

16 

2  25-2  50 

Do 

1.95 

.13 

2  50-2  75 

20 

1  50-1  75 

Do 

.16 

2  25-2  50 

13 

2  50-2  75 

Aggregations(illustrating  stream  trac- 
tion): 
Gravel  

.042 

2.25-2.50 

Do 

.021 

1  25-1  SO 

Sand 

0      -1  00 

WORK  OF  OVERSTROM  AND  BLUE. 

Certain  experimental  work  on  the  capacity  of 
currents  for  flume  traction  has  for  its  specific 
purpose  the  determination  of  dimensions  for 
launders,  the  flumes  in  which  pulverized  ore  is 
conveyed.  R.  H.  Richards's  "Ore  dressing"  4 
contains  an  abstract  of  results  obtained  by  G. 
A.  Overstrom,  accompanied  by  the  statement 
that  the  experimental  data  are  extensive  but  as 
yet  unpublished.  The  troughs  employed  were 
flat-bottomed  and  probably  of  wood.  For  each 
slope,  width  of  flume,  and  grade  of  transported 
material  he  found  (1)  that  the  duty  of  water 
varies  with  the  discharge  and  that  some  par- 
ticular discharge  is  associated  with  a  maximum 
duty,  so  as  to  be  the  most  economical;  (2)  that 

<  Vol.  3,  pp.  1592-1594,  1909. 


FLUME   TRACTION. 


217 


the  most  economical  discharge  is  sensibly  pro- 
portional to  the  width,  so  that  for  each  slope 
and  grade  of  material  there  is  a  particular  dis- 
charge per  unit  of  width  giving  a  maximum 
duty;  and  (3)  that  the  most  economical  dis- 
charge is  greater  for  low  slopes  than  for  high. 

The  first  of  these  results  is  in  fair  accord  with 
our  own.  Five  of  the  nine  values  of  the  expo- 
nent o  in  Table  72  are  loss  than  unity.  For 
the  corresponding  series  the  values  of  ia  range 
both  below  and  above  unity  and  the  corre- 
sponding values  of  the  variable  exponent  for 
duty  in  relation  to  discharge,  i3—  1,  range  below 
and  above  zero.  The  value  zero  evidently  cor- 
responds to  a  maximum  value  of  duty.  His 
third  result  is  in  strict  accordance  with  ours; 
his  second  can  not  be  compared  without  fuller 
details. 

A  diagram  exhibiting  his  determinations  for 
the  traction  of  crushed  quartz  sized  by  40-mesh 
and  150-mesh  sieves  shows  for  different  dis- 
charges per  unit  width  the  variation  of  duty 
with  slope.  For  the  larger  discharges  duty 
varies  as  the  first  power  of  slope ;  for  the  small- 
est discharge  with  the  second  power.  This  cor- 
responds to  a  variation  of  capacity  with  the 
second  to  third  power  of  slope.  Our  most 
available  data  for  comparison  are  those  of 
grade  (C),  the  capacity  for  which  varies  with 
the  1.66  power  of  slope  (Table  70).  As  grade 
(C)  was  separated  by  30-mesh  and  40-mesh 
sieves,  it  is  considerably  coarser  than  the 
crushed  quartz,  and,  being  stream  worn,  it  is 
less  angular.  The  marked  difference  in  the 
observed  laws  of  variation  is  evidently  suscepti- 
ble of  more  than  one  interpretation,  but  it  is 
thought  to  be  connected  with  difference  in 
fineness,  as  more  fully  stated  on  a  following 
page. 

F.  K.  Blue  l  made  a  series  of  experiments  in 
which  the  trough  was  of  sheet  iron,  50  feet  long, 
5  inches  deep,  and  4  inches  wide,  the  bottom 
being  semicylindric  with  2-inch  radius.  It  was 
so  mounted  that  it  could  be  set  to  any  slope 
up  to  12  per  cent.  Two  materials  were  used 
as  load,  the  first  a  beach  sand  of  60-mesh  aver- 
age fineness,  the  other  a  sharp  quartz  sand  of 
about  80-mesh  fineness,  containing  about  10 
per  cent  of  slime  from  a  stamp  mill.  With  each 
material  the  discharge  and  load  were  varied; 
and  for  each  combination  of  discharge  and  load 


i  Eng.  and  Min.  Jour.,  vol.  84,  pp.  530-539, 1907. 


the  slope  was  adjusted  to  competence,  and 
mean  velocity  was  determined  by  means  of  a 
measurement  of  depth.  Discharge  and  load 
were  not  measured  directly  but  in  certain  com- 
binations. Instead  of  discharge,  the  total  vol- 
ume of  water  and  load  was  measured.  This 
quantity  was  used  chiefly  in  the  computation 
of  mean  velocity,  for  which  purpose  it  is  better 
fitted  than  is  discharge  alone.  Load  was  meas- 
ured as  a  volume,  the  volume  of  the  transported 
material  as  collected  in  a  settling  tank,  and  is 
reported  only  through  a  ratio,  q,  which  is  the 
quotient  of  the  volume  of  load  by  the  volume 
of  discharge  plus  load.  This  is  essentially  a 

duty  but  differs  materially  from  duty  (  U=  £.  J 

as  defined  in  the  present  report.  Representing 
by  W  the  weight  in  grams  of  a  cubic  foot  of 
debris,  including  voids,  and  by  v  the  percentage 
of  voids,  it  follows  from  the  definitions  that 

U  qW 

2=  w  +  m=v)'  and      =  1-gd-t;) 

From  the  discussion  of  his  data  Blue  finds 
(1)  that  q  varies  as  the  square  of  the  slope 
and  (2)  that  it  varies  as  the  sixth  power  of  the 
mean  velocity.  He  does  not  specifically  con- 
sider the  relation  of  q  to  discharge,  but  exami- 
nation of  his  tabulated  data  shows  that  q  is 
but  slightly  sensitive  to  variations  of  discharge 
plus  load. 

As  Blue's  coarsest  material,  the  beach  sand, 
has  approximately  the  fineness  of  our  grade  (A), 
while  the  finest  we  treated  in  flume  traction  is 
of  grade  (C),  the  most  definite  comparison  of 
results  can  not  be  made,  but  there  is  neverthe- 
less interest  in  such  comparison  as  is  possible. 
Computing  values  of  £7  from  his  data  for  beach 
sand,  and  plotting  them  in  relation  to  slope,  I 
obtained  UxS2-02  This  gives  for  capacity  and 
slope,  C<xS3M;  and  the  exponent  3.02  may  be 
compared  with  values  of  7t  in  Table  70,  for  the 
smoothest  kind  of  trough  bed.  The  exponent 
for  grade  (C)  is  1.66,  and  the  exponent  has  a 
minimum  value  of  1.30  for  grade  (G).  In 
accordance  with  the  generalization  (p.  208)  that 
the  sensitiveness  of  capacity  to  slope  increases 
from  a  minimum  toward  both  coarse  grades  and 
fine,  we  should  expect  for  grade  (A)  an  index 
of  sensitiveness  materially  greater  than  1.66. 
The  data  furnished  by  Blue  thus  tend  to  sup- 
port the  generalization,  and  additional  support 


218 


TEANSPORTATION    OP    DEBEIS   BY    RUNNING    WATER. 


is  given  by  Overstrain's  data,  above  cited. 
Such  an  inference  is  qualified  as  to  Blue's  data 
by  the  possibility  that  the  exponent  is  affected 
in  material  degree  by  the  form  of  the  cross 
section  of  the  trough. 

Blue's  data  on  velocity  are  of  such  character 
as  to  warrant  a  fuller  discussion  than  he  gives. 
His  comparison  with  duty  (q)  is  made  without 
regard  to  the  accessory  conditions  of  discharge, 
depth,  and  slope,  but  we  have  seen  that  in 
stream  traction  these  conditions  materially  af- 
fect the  control  of  capacity  and  duty  by  mean 
velocity.  It  is  not  practicable  so  to  arrange 
his  data  as  to  obtain  results  for  the  condition 
that  discharge,  depth,  or  slope  is  constant,  but 
moderate  approximations  to  such  conditions 
may  be  obtained  by  grouping.  I  have  divided 
his  observations  on  the  traction  of  beach  sand 
into  three  groups — first,  with  reference  to  dis- 
charge (+load);  second,  with  reference  to 
depth;  and,  third,  with  reference  to  slope — 
and  for  each  group  have  computed  n  on  the 
assumption  that  capacity  varies  with  the  nth 
power  of  mean  velocity.  The  resulting  values 
are  given  in  Table  79,  and  with  them  are  placed 
the  average  of  the  corresponding  exponents  for 
stream  traction,  derived  under  the  several  con- 


ditions of  constant  discharge,  constant  depth, 
and  constant  slope  (Table  53). 

It  appears  (1)  that  in  flume  traction  capacity 
is  much  more  sensitive  to  variation  of  mean  ve- 
locity than  in  stream  traction;  and  (2)  that  in 
flume  traction,  as  in  stream  traction,  the  order 
of  sensitiveness  as  related  to  conditions  is 
highest  for  constant  discharge,  intermediate  for 
constant  depth,  and  lowest  for  constant  slope. 
In  flume  traction  the  sensitiveness  appears  to 
vary  directly  with  discharge  and  depth  and 
inversely  with  slope,  while  in  stream  traction 
it  was  found  to  vary  inversely  with  discharge 
and  slope,  the  variation  with  depth  being  in- 
determinate. 

TABLE  7Q.^Value  of  n  in  C<x  Vmn,  based  on  Blue's 
experiments  on  flume  traction  of  beach  sand. 


Value  of  n  under  condi- 
tion   that    small 
range  is  given  to— 

Q 

i 

a 

The  values  of  Q,  d,  or  S  heing  relatively- 

5.7 
6.2 

7.5 

5.7 
5.0 
7.0 

3.6 
5.0 
3.3 

Intermediate  

Mean 

6.5 

5.9 

4.0 

Comparative  values  of  Ty  from  Table  53  

3.98 

3.68 

3.21 

CHAPTER  XIII.— APPLICATION  TO  NATURAL  STREAMS. 


INTRODUCTION. 

The  flow  of  a  river  is  a  complex  phenomenon. 
The  transportation  of  debris  by  it  involves 
intricate  reactions.  The  quantity  of  debris 
transported  depends  on  a  variety  of  conditions, 
and  these  conditions  interact  one  on  another. 
Direct  observation  of  what  takes  place  at  the 
base  of  the  current  is  so  difficult  that  the  body 
of  information  thus  obtained  is  small.  In  the 
work  of  the  Berkeley  laboratory  the  attempt 
was  made  to  study  the  influence  of  each  con- 
dition separately,  and  to  that  end  all  the  con- 
ditions were  subjected  to  control.  This  in- 
volved the  substitution  of  the  artificial  for  the 
natural;  and  while  the  principles  discovered  are 
such  as  must  enter  into  the  work  of  natural 
streams,  their  combinations  there  are  different 
from  the  combinations  of  the  laboratory.  It  is 
the  province  of  the  present  chapter  to  consider 
the  differences  between  the  laboratory  streams 
and  natural  streams,  and  in  view  of  those 
differences  the  applicability  of  the  laboratory 
results  to  problems  connected  with  natural 
streams. 

FEATURES  DISTINGUISHING  NATURAL 
STREAMS. 

KINDS    OF    STREAMS. 

Classification  necessarily  involves  a  purpose, 
or  point  of  view,  and  there  are  in  general  as 
many  scientific,  or  natural,  or  otherwise  com- 
mendable classifications  as  there  are  functions 
to  be  subserved.  The  classification  of  streams 
here  given  has  no  other  purpose  than  to  afford 
a  terminology  convenient  to  the  subject  of 
debris  transportation. 

When  the  debris  supplied  to  a  stream  is  less 
than  its  capacity  the  stream  erodes  its  bed,  and 
if  the  condition  is  other  than  temporary  the 
current  reaches  bedrock.  The  dragging  of  the 
load  over  the  rock  wears,  or  abrades,  or  cor- 
rades  it.  When  the  supply  of  debris  equals  or 
exceeds  the  capacity  of  the  stream  bedrock  is 
not  reached  by  the  current,  but  the  stream  bed 
is  constituted  wholly  of  debris.  Some  streams 
with  beds  of  debris  have  channel  walls  of  rock, 


which  rigidly  limit  their  width  and  otherwise 
restrain  their  development.  Most  streams  with 
beds  of  d6bris  ha,~ve  one  or  both  banks  of  pre- 
viously deposited  debris  or  alluvium,  and  these 
streams  are  able  to  shift  their  courses  by 
eroding  their  banks.  The  several  conditions 
thus  outlined  will  be  indicated  by  speaking  of 
streams  as  corroding,  or  rock-walled,  or  alluvial. 
In  strictness,  these  terms  apply  to  local  phases 
of  stream  habit  rather  than  to  entire  streams. 
Most  rivers  and  many  creeks  are  corrading 
streams  in  parts  of  their  courses  and  alluvial 
in  other  parts. 

Whenever  and  wherever  a  stream's  capacity 
is  overtaxed  by  the  supply  of  debris  brought 
from  points  above  a  deposit  is  made,  building 
up  the  bed.  If  the  supply  is  less  than  the 
capacity,  and  if  the  bed  is  of  debris,  erosion 
results.  Through  these  processes  streams 
adjust  their  profiles  to  their  supplies  of  d6bris. 
The  process  of  adjustment 'is  called  gradation; 
a.  stream  which  builds  up  its  bed  is  said  to 
aggrade  and  one  which  reduces  it  is  said  to 
degrade. 

An  alluvial  stream  is  usually  an  aggrading 
stream  also ;  and  when  that  is  the  case  it  is  bor- 
dered by  an  alluvial  plain,  called  a  flood  plain, 
over  which  the  water  spreads  in  time  of  flood. 

If  the  general  slope  descended  by  an  alluvial 
stream  is  relatively  steep,  its  course  is  relatively 
direct  and  the  bends  to  right  and  left  are  of 
small  angular  amount.  If  the  general  slope  is 
relatively  gentle,  the  stream  winds  in  an  intri- 
cate manner;  part  of  its  course  may  be  in  direc- 
tions opposite  to  the  general  course,  and  some 
of  its  curves  may  swing  through  180°  or  more. 
This  distinction  is  embodied  in  the  terms  direct 
alluvial  stream  and  meandering  stream.  The 
particular  magnitude  of  general  slope  by  which 
the  two  classes  are  separated  is  greater  for  small 
streams  than  for  large.  Because  fineness  is  one 
of  the  conditions  determining  the  general  slope 
of  an  alluvial  plain,  and  because  the  gentler 
slopes  go  with  the  finer  alluvium,  it  is  true  in 
the  main  that  meandering  streams  are  associ- 
ated with  fine  alluvium. 

219 


220 


TRANSPORTATION    OF    DEBRIS   BY    RUNNING    WATER. 


FEATURES     CONNECTED     WITH     CURVATURE     OF 
CHANNEL. 

As  nearly  all  the  laboratory  experiments  were 
performed  with  straight  channels,  and  as  all 
natural  channels  are  more  or  less  curved,  the 
features  resulting  from  curvature  constitute 
differences  of  which  account  must  be  taken  in 
applying  laboratory  results.  Some  of  these 
differences  have  been  mentioned  in  connection 
with  the  short  series  of  experiments  with 
curved  and  bent  channels,  but  a  fuller  account 
is  desirable. 

In  a  straight  channel  the  current  is  swifter 
near  the  middle  than  near  the  sides  and  is 
swifter  above  mid-depth  than  below.  On 
arriving  at  a  bend  the  whole  stream  resists 
change  of  course,  but  the  resistance  is  more 
effective  for  the  swifter  parts  of  the  stream  than 
for  the  slower.  The  upper  central  part  is  de- 
flected least  and  projects  itself  against  the  outer 
bank.  In  so  doing  it  displaces  the  slow-flowing 
water  previously  near  that  bank,  and  that  water 
descends  obliquely.  The  descending  water  dis- 
places in  turn  the  slow-flowing  lower  water, 
which  is  crowded  toward  the  inner  bank,  while 
the  water  previously  near  that  bank  moves 
toward  the  middle  as  an  upper  layer.  One 
general  result  is  a  twisting  movement,  the  up- 
per parts  of  the  current  tending  toward  the 
outer  bank  and  the  lower  toward  the  inner.1 
Another  result  is  that  the  swiftest  current  is  no 
longer  medial,  but  is  near  the  outer  or  concave 
bank.  Connected  with  these  two  is  a  gradation 
of  velocities  across  the  bottom,  the  greater  ve- 
locities being  near  the  outer  bank.  The  bed 
velocities  near  the  outer  bank  are  not  only 
much  greater  than  those  near  the  inner  bank, 
but  they  are  greater  than  any  bed  velocities  in 
a  relatively  straight  part  of  the  stream.  They 
have  therefore  greater  capacity  for  traction, 
and  by  increasing  the  tractional  load  they  erode 
until  an  equilibrium  is  attained.  On  the  other 
hand,  the  currents  which,  crossing  the  bed  ob- 
liquely, approach  the  inner  bank  are  slackening 
currents,  and  they  deposit  what  they  can  no 
longer  carry. 

It  results  that  the  cross  section  on  a  curve 
is  asymmetric,  the  greatest  depth  being  near 

i  The  system  of  movements  here  described  has  been  observed  by 
many  students  of  rivers.  They  were  demonstrated  by  the  aid  of  a  model 
channel  by  J.  Thomson,  in  connection  with  an  explanation  which  dif- 
fers somewhat  from  that  of  the  present  text.  See  Roy.  Soc.  London 
Proc.,  pp.  5-8,  1876,  and  35C-357, 1877;  also  Inst.  Mech.  ICng.  Proc.,  pp. 
455-400,  1879. 


the  outer  bank.  As  the  winding  stream 
changes  the  direction  of  its  curvature  from  one 
side  to  the  other,  the  twisting  system  of  current 
filaments  is  reversed,  and  with  it  the  system  of 
depths,  but  the  process  of  change  includes  a 
phase  with  more  equable  distribution  of  veloci- 
ties, and  this  phase  produces  a  shoal  separating 
the  two  deeps.  The  shoal  does  not  cross  the 
channel  in  a  direction  at  right  angles  to  its 
sides  but  is  somewhat  oblique  in  position, 
tending  to  run  from  the  inner  bank  of  one 
curve  to  the  inner  bank  of  the  other.  In 
meandering  streams  it  is  usually  narrow  and 
is  appropriately  called  a  bar.  In  direct  alluvial 
streams,  where  bends  are  apt  to  be  separated 
by  long,  nearly  straight  reaches,  it  is  usually 
broad  and  may  for  a  distance  occupy  the 
entire  width  of  the  channel.  In  navigated 
rivers  the  locality  of  the  bar  is  usually  called  a 
crossing,  being  the  place  where  the  thalweg, 
the  line  of  strongest  current,  and  the  route  of 
travel  cross  from  side  to  side;  and  the  name  is 
often  applied  also  to  the  bar  itself. 

The  twisting  current  attacks  the  outer  bank, 
being  swifter  at  contact  with  that  bank  than 
in  any  other  part  of  the  wetted  perimeter.  If 
the  bank  consists  of  alluvium  there  is  erosion, 
the  amount  being  determined  in  part  by 
resistances  arising  from  roots,  or  adhesion  of 
alluvial  particles,  or  incipient  cementation; 
and  the  eroded  material,  so  far  as  it  joins  the 
bed  load,  helps  to  satisfy  the  bed  current  and 
limit  downward  erosion.  In  alluvial  streams 
the  erosion  from  concave  banks  offsets  the 
deposition  under  convex  banks,  so  that  the 
channel  may  gradually  shift  its  position  without 
change  of  sectional  area. 

The  sectional  area  may  be  either  greater  or 
less  at  a  curve  than  on  a  reach,  but  the  differ- 
ences are  normally  1  of  small  amount.  There- 
fore the  mean  velocity  does  not  vary  greatly. 
The  current  in  a  curved  channel,  as  compared 
to  that  in  a  straight  channel,  is  characterized 
by  diversity.  Its  bed  velocities  are  both 
higher  and  lower,  and  the  same  is  true  of 
velocities  along  the  banks.  This  diversity  is 
favorable  to  traction,  because  capacity  for 
traction  varies  with  a  high  power  of  velocity; 
but  the  advantage  to  traction  is  partly  offset 
by  the  fact  that  increase  of  velocity  affects  a 
smaller  portion  of  the  wetted  perimeter  than 

'That  is  to  say,  they  are  of  small  amount  when  the  system  of  depths 
is  adjusted  to  the  discharge,  as  explained  on  a  later  page. 


APPLICATION    TO    NATURAL  STREAMS. 


221 


is  affected  by  redaction  of  velocity.  There  is 
also  diversity  in  the  directions  followed  by 
elements  of  current,  and  this  diversity  includes 
not  only  the  twisting  movement  but  various 
minor  eddies  and  swirls.  Diversified  move- 
ments, by  including  upward  movements,  pro- 
mote suspension,  and  in  conjunction  witb 
diversified  velocities  they  modify  the  partition 
of  the  load  between  traction  and  suspension. 
On  the  whole,  suspension  claims  more  in  a 
diversified  current,  but  it  is  also  true  that  the 
line  of  separation  between  suspension  and 
traction  shifts  to  and  fro  in  such  a  current. 
Much  debris  which  is  suspended  in  the  swift 
water  under  the  concave  bank  joins  the  bed 
load  in  passing  the  shoal  between  deeps,  and 
the  suspended  load  is  still  more  restricted  in 
passing  the  shoal  of  the  convex  bank.  Deposi- 
tion on  the  latter  shoal  includes  both  tractional 
and  suspensional  materials. 

FEATURES    CONNECTED    WITH   DIVERSITY   OF 
DISCHARGE. 

All  streams  vary  in  volume  from  season  to 
season  and  from  year  to  year.  In  a  stream  fed 
by  springs  the  changes  may  be  slight.  At  the 
opposite  extreme  are  creeks  and  even  rivers 
which  exist  only  during  storms.  In  most  large 
streams  the  discharge  at  flood  stage  is  many 
times  greater  than  at  low  stage.  Usually  flood 
stages  continue  only  for  brief  periods  and  in  the 
aggregate  occupy  but  a  small  fraction  of  the 
year. 

It  is  broadly  true  that  streams  give  shape  to 
their  own  channels,  and  among  alluvial  streams 
there  are  few  exceptions.  It  is  broadly  true 
also  that  the  shapes  of  channels,  including  cross 
sections  and  plans,  are  the  same  for  large 
streams  as  for  small.  But  the  large  stream 
requires  and  develops  a  larger  channel — broader, 
deeper,  and  winding  in  larger  curves.  Through 
variation  of -discharge  the  same  stream  is  alter- 
nately large  and  small,  so  that  its  needs  are 
different  at  different  stages.  At  each  stage  it 
tends  to  fit  its  channel  to  the  needs  of  the  par- 
ticular discharge.  The  formative  forces  resid- 
ing in  the  current  are  so  much  stronger  with 
large  discharge  than  with  small  that  the  greater 
features  of  channel  are  adjusted  to  large  dis- 
charge, and  this  despite  the  fact  that  floods 
are  of  brief  duration.  The  feebler  forces  of 
smaller  discharges  modify  the  flood-made  forms 
but  do  not  succeed  in  completing  their  work  of 


adjustment  before  it  is  interrupted  by  another 
flood.  The  deeps  of  high  stage  are  pools  at 
low  stage  and  have  currents  too  feeble  for  trac- 
tion. As  the  reduced  stream  passes  from  pool 
to  pool  it  crosses  the  shoal  formed  at  high 
stage  with  quickened  current.  The  velocities 
are  still  diversified,  but  the  greater  and  smaller 
velocities  have  exchanged  places.  The  slope  of 
water  surface  is  more  diversified  than  at  high 
stage,  being  lower  at  the  pools  and  higher  be- 
tween them.  Traction  is  restricted  to  the 
shoals,  and  the  loads  are  small.  The  load  at 
each  shoal  is  obtained  from  the  shoal  itself  and 
is  deposited  in  the  next  pool,  and  in  this  way 
shallow  channels  are  developed  from  pool  to 
pool. 

In  contrasting  the  features  of  high  and  low 
stages,  it  has  been  convenient  to  use  the  terms 
as  if  high  stage  and  low  stage  were  specific  and 
definite  phases  of  stream  activity,  thereby 
ignoring  the  actual  diversity  in  fluctuations  of 
discharge.  Floods  are  of  all  magnitudes,  and 
each  flood  presents  not  only  a  maximum  dis- 
charge but  a  continuous  series  of  changing  dis- 
charges. At  each  instant  the  stream  contains 
a  system  of  currents  of  which  the  details 
depend  not  only  on  the  discharge  but  on  the 
shapes  of  channel  created  by  the  work  of  pre- 
vious discharges.  So  long  as  the  discharge 
continues,  its  currents  are  eroding  and  deposit- 
ing in  such  way  as  to  remodel  the  channel  for 
its  own  needs,  and  so  long  as  the  work  of 
remodeling  continues  the  loads  and  capacities 
at  different  cross  sections  are  different. 

With  the  changes  in  the  values  and  distri- 
bution of  velocities  go  changes  in  those  values 
of  competent  fineness  which  on  one  side  limit 
traction  and  on  the  other  separate  traction 
from  suspension.  With  maximum  discharge 
all  the  coarser  grades  of  d6bris  within  the 
domain  of  the  stream  are  in  transit  along  the 
path  of  highest  activity,  and  that  path  in- 
cludes the  deeps  and  the  intervening  shoals. 
With  lessening  discharge  the  coarsest  material 
stops,  but  it  stops  chiefly  in  the  deeps,  because 
the  change  in  bed  velocity  is  there  greatest. 
At  the  same  time  the  coarsest  of  the  suspended 
load  escapes  from  the  body  of  the  stream  and 
joins  the  bed  load.  By  this  double  change  the 
mean  fineness  of  the  tractional  load  is  increased, 
and  so  also  is  the  mean  fineness  of  the  suspended 
load.  With  continued  reduction  of  discharge 
the  tractional  load  in  the  deeps  becomes  gradu- 


222 


TBANSPOKTATION   OF   DEBRIS  BY   RUNNING   WATEE. 


ally  finer  and  at  last  ceases  to  move,  while  the 
graduated  deposit  caused  by  its  arrest  receives 
a  final  contribution  from  the  suspended  load. 
The  tractional  load  on  the  shoals  changes  less 
in  mean  fineness  and  may  cease  to  change 
altogether  when  the  supply  from  the  deep  is 
cut  off.  It  is  then  derived  wholly  from  the 
subjacent  bed  and  is  greatly  reduced  in  quan- 
tity. Soon  the  derivation  becomes  selective, 
the  finer  part  being  carried  on  while  the  coarser 
remains,  with  the  result  that  the  shallow  chan- 
nels on  the  bars  come  to  be  paved  with  par- 
ticles which  the  enfeebled  currents  can  not 
move. 

If  the  section  of  the  alluvium  underlying  a 
shoal  be  afterwards  exposed,  it  is  seen  to  be  in 
the  main  heterogeneous  but  veneered  at  the 
top  by  a  layer  of  its  coarser  particles.  The 
typical  section  of  a  deposit  in  a  deep  shows  the 
coarsest  de'bris  below  and  the  finest  at  top, 
with  a  gradual  change. 

With  the  return  of  large  discharge  the  model- 
ing work  of  smaller  discharges  is  rapidly  ob- 
literated, and  the  debris  deposited  in  the  pools 
rejoins  the  tractional  and  suspended  loads. 

SECTIONS    OF    CHANNEL. 

Rock-walled  channels  result  from  the  aggra- 
dation of  corraded  channels.  Often  they  are 
recurrent  temporary  conditions  of  corraded 
channels.  Their  widths  have  been  developed 
in  connection  with  the  work  of  conasion  and 
are  less  than  the  widths  of  alluvial  streams.  In 
the  fact  that  their  sides  are  immobile  they 
lesemble  the  laboratory  channels,  and  their 
types  of  cross  section  are  illustrated  by  the 
experiments  with  crooked  channels.  The 
channels  of  all  alluvial  streams  are  strongly 
asymmetric  at  the  bends,  and  in  the  meander- 
ing streams  the  bends  constitute  the  greater 
part  of  the  course.  Departure  from  symmetry- 
is  less  pronounced  in  the  reaches  of  direct 
alluvial  streams,  but  even  there  a  close  approxi- 
mation to  symmetry  is  exceptionr.l. 

Alluvial  streams  tend  to  broaden  their 
channels  by  eroding  one  or  both  banks.  The 
influence  of  vegetation  opposes  this  tendency. 
Often,  the  erosion  of  the  bank  exposes  roots, 
and  some  trees  extend  rootlets  into  the  water. 
At  low  stages  the  bared  parts  of  the  flood 
channel  are  occupied  by  young  plants.  In 
these  ways  vegetation  creates  obstacles  which 
retard  the  current  at  its  contact  with  the  bank 


and  thus  oppose  erosion.  If  the  current  is 
strong  erosion  is  merely  retarded,  not  pre- 
vented; if  the  current  is  weak  deposition  may 
be  induced.  As  a  meandering  stream  en- 
croaches on  its  concave  bank,  the  convex  bank 
encroaches  on  the  stream,  and  channel  width 
is  maintained.  A  large  stream  is  less  affected 
than  a  small  stream  by  the  opposition  of 
vegetation  and  maintains  a  channel  of  rela- 
tively small  form  ratio. 

Some  streams  aggrade  so  rapidly  that  vegeta- 
tion does  not  secure  a  foothold.  By  erosion 
of  its  banks  such  a  stream  broadens  its  channel 
and  reduces  its  depth  until  the  slackened 
current  clogs  itself  by  deposition  of  its  load. 
The  built-up  bed  becomes  higher  than  the 
adjacent  alluvial  plain,  and  the  stream  takes  a 
new  course.  Before  the  assumption  of  the 
new  course  the  banks  are  overtopped  by 
shallow  distributaries  which  deposit  their  loads 
on  the  banks,  thus  building  them  up,  until  the 
stream  is  made  to  flow  on  a  sort  of  elevated 
conduit;  and  when  the  main  body  of  water  at 
last  leaves  this  pathway,  it  is  apt  to  start  its 
new  course  with  a  steepened  slope  and  scour 
for  itself  a  relatively  narrow  channel. 

The  building  up  of  the  bank  by  deposition 
from  overflow  is  more  pronounced  in  the  pres- 
ence of  vegetation.  The  ridge  thus  created  is 
called  a  natural  levee.  Its  crest  separates  the 
channel  from  the  flood  plain  and  delimits  at 
flood  stage  two  provinces  in  which  the  condi- 
tions affecting  transportation  are  strongly  con- 
trasted. In  both  provinces  the  general  slope 
of  the  water  surface  is  the  same,  but  the  broad 
sheet  covering  the  plain  has  so  little  depth  that 
its  currents  are  sluggish.  Between  the  banks 
are  the  normal  channel  depths  and  currents, 
and  transportation  is  active,  alike  by  traction 
and  suspension.  Beyond  them  transportation 
is  effected  almost  wholly  by  suspension,  and 
the  coarser  particles  of  the  suspended  load  are 
deposited.  As  the  flood  subsides  the  lateral 
sheets  of  water  are  returned  to  the  main  chan- 
nel by  a  draining  process  which  involves  the 
making  and  maintenance  of  small  channels 
within  the  plain. 

When  the  channel  of  a  river  is  fully  adjusted 
to  the  discharge  the  same  load  is  transported 
through  each  section.  All  sections  are  then 
equally  adapted  to  transportation,  though  in 
different  ways.  The  most  symmetric  has  a 
wide  space  at  the  bottom  devoted  to  traction. 


APPLICATION   TO    NATURAL  STREAMS. 


223 


The  least  symjnetric  has  a  relatively  narrow 
tractional  space,  but  traction  is  there  relatively 
active.  The  partition  of  the  load  between 
traction  and  suspension  is  not  the  same  for  the 
two  sections,  the  tractional  load  having  the 
greater  range  in  the  symmetric  section  and  the 
suspended  load  in  the  asymmetric. 

There  is  reason  to  believe  that  the  sectional 
area  is  about  the  same  in  different  parts  of  an 
adjusted  channel.  At  low  stages,  when  form 
is  least  adjusted  to  discharge,  the  sectional  area 
is  much  larger  for  the  asymmetric  sections. 
At  higher  stages  the  contrast  is  less,  and  the 
greater  area  may  be  associated  with  either  type 
of  section.  It  is  also  true,  if  attention  be  re- 
stricted to  the  channel  proper  and  the  expan- 
sions over  flood  plains  be  excluded,  that  the 
variations  in  width  from  point  to  point  of  an 
adjusted  channel  are  not  of  large  amount.  If 
it  were  strictly  true  that  both  sectional  areas 
and  widths  are  equal  in  different  parts  of  an 
adjusted  channel,  it  would  follow  (1)  that  mean 
depths  are  equal,  and  (2)  that  form  ratios  are 
equal,  provided  form  ratio  be  defined  as  the 
ratio  of  mean  depth  to  width.  Such  a  generali- 
zation, while  crude  and  doubtless  subject  to 
important  qualification,  nevertheless  warrants 
the  selection  of  mean  depth  rather  than  maxi- 
mum depth  as  th«e  quantity  to  be  used  in  ap- 
plying the  conception  of  form  ratio  to  rivers. 

Assuming  the  generalization  as  an  approxi- 
mation to  the  actual  fact  and  connecting  with 
it  the  fact  that  all  sections  of  an  adjusted  river 
are  equally  efficient  for  transportation,  we  are 
able  to  make  a  general  application  .of  the 
laboratory  results  on  optimum  form  ratio  to 
rivers.  The  ratio  of  mean  depth  to  width  in 
alluvial  rivers,  as  a  class,  is  very  much  smaller 
than  in  the  laboratory  examples  by  means  of 
w.hich  the  optimum  ratio  was  discussed  in 
Chapter  IV.  It  is  so  much  smaller  that  the 
range  of  form  ratio  for  alluvial  rivers  overlaps 
but  slightly  the  range  observed  in  the  labora- 
tory. This  disparity  indicates,  though  without 
demonstrating,  that  the  form  ratios  of  the 
rivers  are  less  than  the  optimum,  and  that  their 
tractional  capacities  would  be  greater  if  they 
were  narrower  and  deeper.  As  the  optimum 
ratio  is  the  one  which  enables  a  stream  to 
transport  its  load  with  the  least  expenditure  of 
head,  it  is  probable  that  the  slopes  of  most 
alluvial  rivers  can  be  lessened  by  artificially 
reducing  their  widths. 


THE    SUSPENDED  .LOAD. 

In  speaking  above  of  the  transfer  of  load  from 
traction  to  suspension  no  consideration  was 
given  to  capacity  for  suspension.  Certain  stu- 
dents of  rivers,  comparing  discharges  or  veloci- 
ties with  the  percentage  of  suspended  material 
and  finding  a  rough  correspondence,  have  in- 
ferred that  suspended  load  is  a  function  of 
velocity;  others,  giving  attention  to  conspicu- 
ous examples  of  noncorrespondence,  have  in- 
ferred that  the  suspended  load  depends  only 
on  the  supply  of  suitable  material.  There  is  a 
taeasure  of  truth  in  both  views,  and  their  diver- 
gence is  largely  to  be  explained  by  considera- 
tions connected  with  competence. 

The  subject  is  illustrated  by  observations  on 
the  suspended  load  of  Yuba  River.  Its  water 
was  sampled  during  flood  stages,  in  1879  '  at 
Marysville,  a  load  of  0.35  per  cent,  by  weight, 
being  found  when  the  discharge  was  estimated 
at  26,000  ft.3/sec.,  and  a  load  of  0.42  per  cent 
when  the  discharge  was  18,000  ft.3/sec.  In 
1906,  at  a  time  when  the  discharge  was  esti- 
mated at  33,000  ft.3/sec.,  samples  were  taken 
at  the  same  place  and  the  load  was  found  to 
be  only  0.065  per  cent.  At  the  time  of  the 
earlier  samplings  hydraulic  mining  was  in  full 
operation  in  the  basin  of  the  Yuba,  and  the 
suspended  load  consisted  chiefly  of  clay  com- 
ponents of  the  auriferous  gravels,  artificially  fed 
to  the  stream.  In  1906  there  was  little  hy- 
draulic mining  and  the  suspended  load  con- 
sisted of  material  washed  from  the  surface  of 
the  land  by  rain. 

At  low  stages  in  1906  the  water  at  Marysville 
was  clear,  but  in  1879  a  sampling  when  the 
discharge  was  510  ft.3/sec.  gave  a  load  of  O.S<5 
per  cent.  The  fact  that  the  river's  load  in  1879 
constituted  a  higher  percentage  at  low  stage 
than  during  flood  is  explained  by  the  considera- 
tion that  the  turbid  tributaries  from  the  mines 
were  less  diluted  by  other  tributaries  at  low 
stage  than  at  high. 

If  we  assume,  first,  that  0.065  per  cent,  ob- 
served at  flood  stage  in  1906,  represents  the 
normal  tribute  from  other  sources  than  mining, 
and,  second,  that  the  mines  contributed  the 
same  total  amounts  at  low  and  high  stages  in 
1879  (discharges  being  510  and  18,000  ft.3/sec.), 
and  if  we  base  on  these  assumptions  a  compu- 

i  Manson,  Marsden,  Report  of  determinations  of  sediment  held  in  sus- 
pension, etc.:  California  State  Engineer  Report,  1880,  Appendix  B. 


224 


TRANSPORTATION    OF    DEBRIS    BY    RUNNING    WATER. 


tation  of  the  low  stage  percentage  of  load  in 
1879,  we  obtain  an  estimate  of  12.6  per  cent, 
which  is  14  times  greater  than  the  observed 
load,  0.86  per  cent.  The  discrepancy  is  alto- 
gether too  great  to  be  accounted  for  by  errors 
in  the  explicit  assumptions,  and  its  explana- 
tion involves  the  factor  of  competence.  The 
stronger  currents  of  flood  stages  were  compe- 
tent to  suspend  .heavier  particles  than  were  the 
feeble  currents  of  low  stages,  and  so  a  part  of 
the  load  which  at  high  stages  reached  the  mouth 
of  the  river  at  Marysville  was  at  low  stages 
arrested  on  the  way,  being  deposited  in  the 
low-water  pools. 

In  these  cases  it  is  evident  that  the  range  of 
available  fineness  is  determined,  through  veloc- 
ity, by  discharge,  and  that  the  load  of  debris 
within  the  range  of  adeqiiate  fineness  is  deter- 
mined by  supply.  The  load  appears  to  bear 
no  relation  to  capacity,  and  if  the  term  capacity 
be  used  in  the  broad  sense  of  a  stream's  ability 
to  suspend  material  of  unspecified  fineness,  then 
it  is  undoubtedly  true,  not  merely  of  the  Yuba 
but  of  all  rivers,  that  the  suspended  load  is  less 
than  the  capacity  and  depends  for  its  quantity 
on  supply.  If,  however,  capacity  be  considered 
with  reference  to  particular  degrees  of  fineness, 
the  case  is  somewhat  different,  for  a  stream 
may  carry  a  full  load  of  that  material  for  which 
it  is  barely  competent  and  at  the  same  time 
have  less  than  a  full  load  of  finer  material,  and 
the  matter  is  further  complicated  by  interde- 
pendencies  in  virtue  of  which  each  element  of 
load  tends  to  limit  the  capacity  for  all  other 
elements  of  load. 

To  show  the  basis  for  these  statements  and 
also  to  explain  certain  mutual  relations  between 
traction  and  suspension,  it  is  necessary  to  give 
somewhat  elementary  consideration  to  the  sub- 
ject of  capacity  for  suspension. 

As  already  mentioned  and  implied  in  various 
connections,  the  process  of  suspension  depends 
on  the  diversity  in  direction  of  the  strands  of 
the  current.  If  the  lines  of  flow  were  parallel 
to  the  stream  bed,  as  is  sometimes  assumed  for 
the  sake  of  simplifying  mathematical  discus- 
sions, there  would  be  no  suspension.1  In  the 

i  There  is  a  theory  originating  with  Dupuit  (Etudes  sur  le  mouve- 
ment  des  eaux,  1848)  that  suspension  is  due,  or  might  be  duo  in  the 
ideal  case  of  parallel  flow  lines,  to  reactions  between  solid  particles  and 
contiguous  threads  of  current  having  different  velocities.  Under  the 
postulate  that  the  solid  tends  to  move  faster  than  the  liquid,  it  is  shown 
that  the  path  of  least  resistance  trends  obliquely  toward  the  swifter  of 
adjacent  threads  of  current,  and  therefore  obliquely  upward.  As  this 
theorj  retains  place  in  current  hydraulic  literature,  the  fact  that  it  is 


sinuous  and  swirling  movements  which  charac- 
terize the  flow  of  streams  strands  of  current  are 
continually  passing  upward  and  downward  and 
are  as  continually  dividing  and  blending.  Par- 
ticles of  debris  too  light  to  resist  the  lower  ele- 
ments of  the  current  are  swept  upward  and  are 
retained  in  the  body  of  the  stream  through  a 
process  analogous  to  the  stirring  of  the  domes- 
tic pot.  While  thus  incorporated  they  are 
impelled  downward  by  gravity,  and  all  but  the 

ignored  in  the  text  of  the  present  paper  may  call  for  explanation.  I  do 
not  accept  the  postulate  and  am  of  opinion  also  that  the  reasoning  based 
on  it  ignores  an  essential  factor.  As  a  full  statement  and  discussion  of 
Dupuit's  analysis  would  occupy  mucU  space,  I  will  content  myself 
with  a  statement  of  my  own  view.  A  good  abstract  of  h  is  theory,  by 
E.  II.  Hooker,  may  be  found  in  Am.  Soc.  Civil  Eng.  Trans.,  vol.  38,  pp. 
246-247,  320-322. 

In  various  discussions  of  the  subject  the  velocities  are  treated  as 
"absolute"— that  ts,  they  are  referred  to  the  fixed  walls  and  bed  of  the 
channel.  As  the  only  possible  reactions  between  the  solid  particles  and 
contiguous  water  are  through  relative  velocities,  it  is  better  to  focus 
attention  on  those  by  referring  them  to  the  center  of  the  particle.  Let 
us  assume  that  the  solid  particle  A ,  figure  70,  is  immersed  in  a  current 


FIODBE  70.— Diagram  of  forces. 

of  which  the  parallel  rectilinear  filaments  increase  gradually  in  "abso- 
lute" velocity  from  below  upward,  and  let  us  assume  that  at  some 
instant  it  moves  with  the  velocity  and  direction  of  the  filament  which 
is  at  the  same  level.  Barring  extraneous  forces,  it  will  continue  indefi- 
nitely in  the  same  direction  and  with  the  same  velocity.  The  filament 
above  moves,  with  reference  to  the  particle,  in  a  direction  indicated  by 
the  arrow;  the  filament  below  moves  in  the  opposite  direction.  Their 
relative  velocities  are  the  same,  except  for  a  possible  difference  of  the 
second  order  of  magnitude.  The  two  filaments  tend  to  draw  the  upper 
and  lower  parts  of  the  particle  in  opposite  directions,  and  the  result  Is 
rotation.  This  is  the  only  result  dependent  on  the  fact  that  the  particle 
is  solid.  Now  introduce  the  factor  of  density.  The  particle  is  denser 
than  water.  It  is  also  part  of  a  stream  which  is  flowing,  and  the  impulse 
It  receives  from  gravity  is  greater  than  it  would  receive  if  it  had  the 
density  of  water.  The  component  of  gravity  in  the  direction  of  flow, 
AB,  acting  on  the  excess  of  mass,  draws  the  particle  in  the  direction  of 
flow.  This  component  is  proportional  to  the  slope  of  the  stream,  which 
is  a  small  fraction.  At  the  same  time  the  component  of  gravity  normal 
to  the  direction  of  flow,  A  C,  also  draws  the  particle,  which  is  equally 
free  to  move  through  the  water  in  that  direction.  Its  actual  accelera- 
tion has  the  direction  of  their  resultant,  AO,  which  is  vertically 
downward. 

Dupuit's  postulate  was  suggested  and  supported  by  the  observed 
fact  that  a  body  floating  down  a  stream  moves  faster  than  the  visible 
current.  BeYard  demonstrated  experimentally  that  the  differential 
motion  is  due  to  the  propulsion  of  the  body  by  strands  of  current  below 
the  surface.  See  Annales  des  ponts  et  chauss&s,  6th  ser.,  vol.  12,  pp. 
830-835, 1886. 


APPLICATION    TO    NATURAL  STREAMS. 


225 


very  finest  actually  move  downward  with  refer- 
once  to  the  surrounding  water.  From  time  to 
time  they  may  touch  the  stream  bed,  but  only 
to  be  lifted  again  by  the  next  adequate  rush  of 
water. 

Three  ways  are  known  in  which  the  veloci- 
ties of  a  stream  are  affected  by  the  suspended 
load.  In  the  first  place,  the  load  adds  its  mass 
to  the  mass  of  the  stream,  and  as  the  stream's 
energy  is  proportional  to  the  product  of  mass 
by  slope,  and  the  stream's  velocity  has  its 
source  in  this  energy,  the  addition  tends  to 
increase  velocity.  Second,  the  suspended  par- 
ticles are  continually  impelled  downward  by 
gravity.  Also,  as  the  strands  of  current  con- 
taining them  have  curved  courses,  the  particles 
are  subject  to  tangential  force,  and  because  of 
their  higher  density  this  force  is  greater  than  the 
force  simultaneously  developed  by  the  contain- 
ing water,  so  that  they  are  impelled  through  the 
water.  Motion  through  the  water,  caused  by 
these  two  forces,  involves  work;  and  this  work 
is  a  direct  consequence  of  suspension.  The 
energy  expended  is  potential  energy,  or  energy 
of  position,  given  to  the  particles  by  the  flo'w- 
ing  water,  and  its  source  is  the  energy  of  flow. 
So  the  work  is  the  measure  of  the  work  of  the 
stream  in  suspending  the  particles.  It  may 
therefore  be  regarded  as  a  tax  on  the  stream's 
energy,  resulting  in  reduction  of  velocity. 
Third,  the  imperfect  liquid  constituted  by  the 
combination  of  water  and  debris  is  more  viscous 
and  therefore  flows  more  slowly  than  the  water 
alone.  The  solid  particles  do  not  partake  of 
the  internal  shearing  involved  in  the  differen- 
tial movements  of  the  current,  and  by  their 
rigidity  they  restrain  the  shearing  of  water  in 
their  immediate  vicinity.  Moreover,  each  par- 
ticle is  surrounded,  through  molecular  forces, 
by  a  sphere  or  shell  of  influence  which  still 
further  interferes  with  the  freedom  of  water 
movement. 

The  relative  importance  of  these  factors 
varies  with  conditions,  and  no  sun  pie  statement 
is  possible  because  the  influence  of  each  factor 
follows  a  law  peculiar  to  itself.  The  most 
important  conditions  affecting  the  influence  of 
the  mass  of  the  load  are  discharge  and  slope, 
while  for  the  work  of  suspension  and  for 
viscosity  the  important  condition  is  the  degree 
of  comminution  of  the  load. 

The  mass  which  the  load  contributes  to  the 
stream  is  equivalent,  in  relation  to  potential 

20921°— No.  86—14 ^15 


energy,  to  an  increase  of  discharge,  and  its 
product  by  the  stream's  slope  is  proportional 
to  potential  energy  in  the  same  sense  in  which 
the  stream's  energy  is  proportional  to  the  pro- 
duct of  discharge  and  slope.  In  a  series  of 
experiments  with  loadless  streams  of  water 
flowing  in  straight  troughs,  the  mean  velocity 
was  found  to  vary  approximately  with  the  0.25 
power  of  the  discharge  and  the  0.3  power  of 
the  slope.  Under  the  particular  conditions  of 
these  experiments 

Vm  =  Q°-2SS°-3x  constant. 
Differentiating  with  reference  to  Q,  we  have 


d  Vm  = 


C0.3 

—  X  constant  . 


The  increment  to  Q  being  interpreted  for  pres- 
ent purposes  as  the  suspended  load,  we  see  that 
the  corresponding  increment  to  mean  velocity 
has  a  magnitude  which  varies  directly  but 
slowly  with  the  slope  and  inversely  but  more 
rapidly  with  the  discharge. 

Each  particle  in  suspension  is  drawn  down- 
ward through  the  surrounding  water  by 
gravity.  It  is  impelled  through  the  water  in 
an  ever-changing  direction  by  tangential  force. 
The  average  speed  of  the  resultant  motions, 
referred  to  the  surrounding  water,  .  is  greater 
than  the  constant  rate  of  descent  the  particle 
would  acquire  if  sinking  in  still  water.  There- 
fore the  work  of  suspension,  measured  by  all 
the  motions  through  the  water,  is  greater  than 
the  work  of  simple  subsidence,  a  quantity  as  to 
which  much  is  known.  The  measure  of  a 
particle's  work  of  subsidence  per  unit  tune  is 
the  product  of  its  mass,  less  the  mass  of  an 
equal  volume  of  water,  by  its  fall  in  unit  time 
by  the  acceleration  of  gravity.  If  we  call  the 
mass  of  the  particle  M  and  its  velocity  of 
subsidence  V,,  and  assume  its  density  to  be 
2.7,  the  measure  of  the  work  of  subsidence  is 


The  coordinate  measure  of  work  for  the 
stream's  flow  is  the  product  of  its  mass  by  its 
fall  in  unit  time  by  the  acceleration  of  gravity, 
and  the  contribution  which  the  particle,  con- 
sidered as  a  part  of  the  stream,  makes  to  the 
work  of  flow  is  therefore  measured  by  the 
product  of  its  mass  by  the  fall  of  the  stream  in 


226 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


unit  time  by  the  acceleration  of  gravity.  As 
the  fall  of  the  stream  in  unit  time  is  equal  to 
the  mean  velocity  multiplied  by  the  slope,  the 
measure  of  the  particle's  work  of  flow  is  Ms  Vmg. 
This  is  a  measure  of  the  particle's  contribution 
to  the  stream's  energy,  while  0.63  MVsg  is  a 
coordinate  measure  of  that  factor  of  its  draft 
on  the  stream's  energy  which  depends  on  the 
direct  action  of  gravity.  The  other  factor  of 
draft,  the  factor  depending  on  tangential 
forces,  varies  with  the  violence  of  vertical 
movements  and  is  of  comparable  importance 
only  in  the  case  of  torrents.  If  we  leave  it  out 
of  account,  the  result  of  the  comparison  is  that 
when  0.63  of  the  rate  at  which  the  particle  is 
pulled  through  the  water  by  gravity  is  greater 
than  the  rate  at  which  it  falls  by  reason  of  the 
general  descent  of  the  stream,  its  tax  on  the 
stream's  energy  is  greater  than  its  contribution 
thereto.  Any  allowance  for  the  neglected 
factor  would  be  equivalent  to  increasing  the 
fraction  0.63. 

To  illustrate  this  relation  by  a  concrete 
example:  Mississippi  River  between  Cairo  and 
its  mouth  has,  at  flood  stage,  such  velocity  and 
slope  that  any  suspended  particle  of  silt  which 
would  sink  in  still  water  faster  than  half  an 
inch  a  minute  retards  the  current  more  through 
the  work  of  suspension  than  it  accelerates  the 
current  through  the  addition  of  its  mass  to  the 
mass  of  the  stream. 

The  velocity  of  subsidence  has  been  elabo- 
rately studied.  So  far  as  river  problems  are 
concerned,  it  depends  chiefly  on  the  size  of  the 
suspended  particles.  For  particles  below  a 
certain  magnitude,  which  is  controlled  in  part 
by  impurities  in  solution,  the  velocity  is  zero. 
Between  two  critical  diameters,  which  for 
quartz  sand  are  about  0.02  and  0.5  millimeter  ' 
(0.00007  and  0.00016  foot)  the  velocity  varies 
with  the  square  of  the  diameter.  Below  the 
lower  diameter  the  variation  is  more  rapid, 
and  above  the  upper  it  is  less  rapid,  becoming 
for  large  particles  as  the  square  root  of  the 
diameter. 

It  follows  that  the  consumption  of  energy 
involved  in  the  suspension  of  the  suspended 
load  is  an  increasing  function  of  the  size  of  the 
particles  into  which  the  load  is  divided;  in 
other  words,  it  is  a  decreasing  function  of  the 
degree  of  comminution.  On  the  other  hand, 

'  Richards,  R.  H.,  Textbook  on  ore  dressing,  pp.  2C2-2C8, 1909. 


the  contribution  which  it  makes  to  energy  by 
adding  its  mass  to  that  of  the  water  is  inde- 
pendent of  the  degree  of  comminution. 

The  viscosity  factor  is  not  easily  compared 
in  a  quantitative  way  with  those  just  consid- 
ered, but  something  may  be  said  of  the  laws 
by  which  it  is  related  to  comminution.  At- 
tending first  to  that  part  which  depends  on 
interference  with  internal  shearing  of  the  water, 
let  us  conceive  of  a  particle  with  center  at  C, 
figure  71,  surrounded  by  water  which  is  sub- 
jected to  uniformly  distributed  shearing  along 
planes  parallel  to  A^AA^BJiB^  the  direction 
of  shearing  being  parallel  to  the  line  ACE. 
Conceive  a  right  cylindroid  figure  tangent  to 
the  particle  and  parallel  to  AB,  its  bases  being 
AJIA2G  and  B1EB2F.  Motions  being  re- 
ferred to  0  as  origin,  the  cylindroid  body  of 
water  would  assume  after  a  time,  but  for  the 
presence  of  the  particle,  the  form  of  the  oblique 
cylindroid  with  bases  A1U1A2G1  and  B1E1B2F1. 


f,    f 

FIGURE  71. — Interference  by  suspended  particle  with  freedom  of  shearing. 

Because  of  the  obstruction  by  the  rigid  par- 
ticle the  simple  shearing  motions  thus  indi- 
cated are  replaced  by  other  motions,  compo- 
nents of  which  are  normal  to  the  shearing 
planes.  It  is  assumed  that  the  sum  of  the 
transverse  elements  of  motion  measures  the 
action  occasioned  by  the  presence  of  the  par- 
ticle. The  actual  movements  caused  in  the 
water  doubtless  affect  regions  within  and  with- 
out the  cylindroid,  but  their  nature  need  not 
be  considered.  The  necessary  transverse  move- 
ments are  equivalent  to  the  transfer  of  a  lunate 
wedge  of  water,  HAJI1A2  to  a  symmetric 
position  on  the  opposite  side  of  the  plane 
AlA2B2Bl  and  a  similar  transfer  of  the  wedge 
FB1FiB2.  Linear  dimensions  of  the  first- 
named  wedge,  in  the  direction  of  rectangular 
coordinates,  are  A^A^,  HA,  and  HII^.  A^A2 
equals  a  diameter  of  the  particle.  IIA  equals 
a  semidiameter.  As  the  angle  HAIIlt  being 
given  by  the  general  amount  of  shearing,  is 
independent  of  the  size  of  the  particle,  IIH, 


APPLICATION    TO    NATURAL   STREAMS. 


227 


equals  AHxt&n  IIAII^  and  is  proportional  to 
a  diameter  of  the  particle.  On  the  assumption 
that  the  diameters  of  the  particle  are  equal  it 
follows  that  the  volume  of  the  wedge  is  pro- 
portional to  that  of  the  particle,  or  to  D3.  The 
distance  of  the  center  of  gravity  of  the  wedge 
from  the  plane  A^A^E^B^,  being  a  linear  di- 
mension of  the  wedge,  is  proportional  to  D,  and 
the  mean  distance  of  transfer,  which  is  double 
that  distance,  is  also  proportional  to  D.  The 
quantity  of  motion  normal  to  the  shearing 
planes,  occasioned  by  the  particle,  is  measured 
by  the  wedge  of  water  times  the  mean  distance 
of  transfer,  and,  the  mass  of  the  wedge  being 
proportional  to  its  volume,  the  quantity  of 
motion  is  proportional  to  D3xD  =  D*.  The 
quantity  of  motion  occasioned  by  the  entire 
suspended  load,  all  its  particles  being  assumed 
to  have  the  same  size,  is  proportional  to  ND4, 
where  N  is  the  number  of  particles ;  and  if  the 
mass  of  the  total  load  remain  unchanged  while 
its  degree  of  comminution  is  varied,  it  is  evident 
that  N  varies  inversely  with  D3.  Therefore  the 
quantity  of  internal  motion  occasioned  by  a 
particular  mass  of  suspended  matter  of  uniform 
grain  is  proportional  (since  D4/D3  =  D)  to  the 
diameter  of  its  particles.  The  fundamental 
assumption  of  the  analysis  is  that  this  motion 
measures  a  resistance  to  the  freedom  of  the 
water  which  is  coordinate  with  viscosity  and 
which  may  for  practical  purposes  be  considered 
as  an  addition  to  the  resistance  arising  from 
the  viscosity  of  the  water. 

That  portion  of  the  viscosity  factor  which 
depends  on  the  molecular  influence  of  the 
particle  outside  its  boundary  is  still  less  sus- 
ceptible of  quantitative  estimate  but  may  yet 
be  discussed  with  reference  to  the  diameter  of 
the  suspended  particle.  In  ignorance  of  the 
exact  nature  of  the  influence  and  also  of  the 
law  by  which  it  diminishes  with  distance  out- 
ward from  the  boundary,  I  assume  arbitrarily 
that  at  all  distances  from  the  particle  less  than 
I  the  freedom  of  water  molecules  is  restricted, 
and  that  the  amount  of  restriction  is  measured 
by  the  volume  of  the  space  in  which  the  restric- 
tion is  experienced.  The  imperfection  of  this 
assumption  will  of  course  affect  any  deduction 

from  it.     That  volume  is  J  TT  (D  +  2l)3  -  ~  r.  D3. 

For  the  entire  suspended  load,  assumeil  to 
consist  of  N  equal  particles,  the  total  volume 


is  N  times  as  great.  For  a  load  of  invariable 
weight  but  variable  comminution,  N<XJ^,  and 
the  total  volume  is  proportional  to 


D3 

or,  introducing  a  constant,  It,  and  reducing,  we 
have 


Volume  = 


Evidently  the  influence  of  this  factor  varies 
inversely  with  D.  When  D  is  very  large  as 
compared  to  2l,  it  approaches  zero;  when  D  is 
small  as  compared  to  2l,  it  is  relatively  very 
great.  It  is  most  sensitive  to  the  control  by  D 
when  the  particles  are  very  small. 

The  two  divisions  of  the  viscosity  factor  vary 
in  their  influence  on  velocity  with  the  comminu- 
tion of  the  load  but  in  opposite  ways,  the 
influence  of  the  first  being  greater  as  the 
particles  are  larger  and  that  of  the  second  as 
they  are  smaller.  The  laws  of  variation  are 
such  that  their  combination  exhibits  a  mini- 
mum— that  is,  for  some  particular  size  of 
particle  the  influence  on  velocity  is  less  than 
for  particles  either  larger  or  smaller. 

Another  mode  of  treating  the  viscosity 
factor  assumes  that,  so  far  as  the  viscosity 
effect  is  concerned,  the  molecular  influence  is 
equivalent  to  an  enlargement  of  each  particle 
to  the  extent  of  I  on  all  sides.  Then,  reasoning 
as  before  with  reference  to  interference  with 
shearing,  we  obtain 


Resistance  oc 


(D  +  21Y 
D3 


This  expression  is  not  only  simple  but  has  the 
advantage  of  giving  definite  indication  of  the 
position  of  the  minimum.  The  resistance  is 
least  when  Z?  =  6Z. 

We  may  now  bring  together  the  qualitative 
results  of  analysis,  and  write 


Vm+ 


This  may  be  read:  The  mean  velocity  (F^,)  of 
a  stream  carrying  a  load  of  suspended  debris 
of  diameter  D  equals  the  mean  velocity  (Fm) 
of  the  same  stream  when  without  load,  plus  a 


228 


TRANSPORTATION    OF   DEBRIS  BY   RUNNING   WATER. 


factor  due  to  the  weight  (W)  of  the  load  and 
varying  directly  with  the  slope  and  inversely 
with  the  discharge,  minus  a  factor  due  to  the 
work  of  suspension  and  varying  directly  with 
the  diameter  of  particles,  minus  a  factor  due 
to  viscosity  and  varying  inversely  with  the 
diameter  if  the  particles  are  minute  and 
directly  with  the  diameter  if  they  are  larger. 

It  was  convenient  in  the  discussion  to  regard 
the  whole  suspended  load  as  of  one  grade,  but 
the  result  may  be  applied  to  any  individual 
particle.  For  the  actual  stream  with  diversi- 
fied load  the  equation  might  be  written 

F™=  Vm+  Wf^S,  $)  - 


.  .(Ill) 

An  investigation  of  the  influence  of  suspended 
matter  on  viscosity  has  recently  been  made  by 
Eugene  0.  Brigham  and  T.  C.  Durham.1  The 
materials  used  by  them  were  powders  so  fine 
that  their  tendency  to  settle  in  the  water  did 
not  interfere  with  the  conduct  of  the  experi- 
ments. The  work  of  suspension  was  therefore 
so  small  a  factor  as  to  be  negligible.  Various 
mixtures  of  water  and  powder  were  allowed  to 
flow  through  a  vertical  capillary  tube,  impelled 
by  their  own  weight,  and  the  time  of  trans- 
mission for  a  determinate  volume  was  noted. 
The  time  for  a  mixture  containing  100  parts 
of  water  to  2.15  parts,  by  weight,  of  clay  was 
found  to  be  15  per  cent  greater  than  the  time 
for  clear  water;  or  the  average  velocity,  within 
a  slender  tube,  was  reduced  15  per  cent  by  the 
addition  of  the  clay.  If  we  assume  the  flow 
lines  to  have  been  parallel,  as  was  probable, 
the  theoretic  increment  of  velocity  due  to  the 
weight  added  by  the  clay  was  1.6  per  cent;  if 
the  flow  lines  were  sinuous,  the  theoretic  incre- 
ment was  less.  The  loss  of  velocity  due  to 
increased  viscosity  was  therefore  somewhat 
greater  than  15  per  cent  and  may  have  been 
as  much  as  16.6  per  cent.  Had  the  experiment 
been  so  arranged  as  to  involve  sinuous  or  tur- 
bulent flow,  it  would  apply  more  cogently  to 
the  phenomena  of  rivers,  but  without  that 
adjustment  it  serves  to  show  that  when  the 
suspended  material  is  exceedingly  fine  the  loss 
of  velocity  through  added  viscosity  is  very 
much  greater  than  the  gain  of  velocity  because 
of  added  weight. 

The  size  of  the  particles  was  not  reported, 
but  the  fact  that  the  rate  of  settling  was  in- 

i  Am.  Chem.  Jour.,  vol.  46,  pp.  278-297,  Mil. 


appreciable  suggests  that  their  size  may  have 
fallen  below  that  corresponding  to  a  minimum 
influence  on  viscosity.  Some  light  is  thrown 
on  this  point  by  the  results  obtained  with  other 
fine  powders.  Interpolation  from  the  reported 
data  gives  the  following  comparative  estimates 
of  the  loss  in  velocity  from  the  suspension  of 
the  same  volumes  of  three  substances:  Infu- 
sorial earth,  3  per  cent;  graphite,  12  per  cent; 
clay,  15  per  cent.  Of  these  the  infusorial  earth 
was  coarsest,  remaining  in  suspension  largely 
because  differing  little  in  density  from  the 
water,  while  the  graphite  was  a  commercial 
variety  said  to  be  permanently  suspended.  An 
independent  determination  by  L.  J.  Briggs  and 
Arthur  Campbell 2  gave  a  loss  in  velocity  of 
7  per  cent,  the  material  being  a  clay  which 
"would  not  remain  in  a  state  of  permanent 
suspension."  The  comparative  data  render  it 
probable  that  the  clay  causing  a  retardation 
of  15  per  cent  was  so  finely  divided  as  to  give 
great  effect  to  the  molecular  forces  of  the  shell 
of  influence. 

The  data  from  infusorial  earth  are  useful  in 
correlating  the  various  factors  which  modify 
the  stream's  velocity.  Assuming  that  the  dia- 
tom tests  composing  the  earth  sample  were 
similar  to  those  figured  in  geologic  textbooks, 
I  have  estimated  the  velocity  of  subsidence  of 
particles  having  the  same  size  but  as  dense  as 
ordinary  river  silt  and  find  that  it  corresponds 
to  the  critical  velocity  computed  for  the  Mis- 
sissippi at  flood  stage.  That  is,  a  suspended 
silt  so  fine  as  to  have  a  large  viscosity  effect,  so 
that  a  charge  of  2.15  per  cent  reduces  velocity 
by  3  per  cent,  is  at  the  same  time  able,  through 
its  work  of  suspension,  to  consume  all  the 
energy  it  contributes  to  the  current  through 
its  addition  of  mass.  A  finer  silt  would  retard 
more  by  increasing  viscosity,  and  a  coarser  silt 
would  retard  more  through  the  work  of  sus- 
pension. 

The  available  data  are  not  fully  demonstra- 
tive, but  they  render  it  highly  probable  that, 
under  all  conditions,  streams  are  retarded  by 
their  suspended  loads.  If  that  be  true,  there 
is  a  capacity  for  suspension  coordinate  with 
capacity  for  traction.  For  each  grade  of  sus- 
pended debris,  and  with  any  particular  slope 
and  discharge,  it  is  possible  by  increasing 
the  load  so  to  retard  the  current  that  it  is 

*  Unpublished;  communicated  in  lettw. 


APPLICATION    TO    NATURAL    STREAMS. 


229 


barely  competent  to  suspend  debris  of  that 
grade,  and  the  stream  is  then  loaded  to  its  full 
capacity. 

In  a  natural  stream  the  suspended  particles 
are  of  many  sizes,  and  all  tend  to  reduce  ve- 
locity. Some  of  the  particles  are  near  the 
limit  of  competence,  and  any  increase  of  load 
will  so  reduce  velocity  that  these  can  no  longer 
be  upheld.  As  to  these  coarsest  particles  the 
stream  is  loaded  to  full  capacity,  but  not  as  to 
finer  material.  The  addition  of  fine  material 
will  cause  the  arrest  of  some  of  the  coarser, 
but  will  increase  the  total  load.  An  exactly 
parallel  statement  may  be  made  as  to  the  trac- 
tional  load. 

To  recur  to  the  laboratory  determinations  of 
retardation  by  suspended  material,  it  is  of  in- 
terest to  note  that  the  loads  tested  in  the  ex- 
periments, while  greater  than  those  ordinarily 
found  in  rivers,  are  representative  of  flood 
conditions  in  the  more  turbid  streams.  The 
load  of  2.15  per  cent,  which  caused  retardations 
of  3  to  15  per  cent,  is  equaled  by  ordinary  floods 
of  the  Colorado  of  the  West  and  is  exceeded 
by  ordinary  floods  of  the  Rio  Grande  and  the 
Pecos.  For  the  Rio  Grande  there  are  several 
records  in  the  neighborhood  of  10  per  cent,1 
and  small  streams  in  arid  lands  are  liable  to 
receive  similar  loads  as  a  result  of  violent  local 
storms.  On  the  other  hand,  the  Mississippi 
near  its  mouth  carries  an  average  load  of  only 
0.07  per  cent,  with  a  recorded  maximum  of 
0.8  per  cent.2 

An  attempt  was  made  to  measure  the  re- 
tardation of  the  current  by  tractional  load.  It 
was  assumed  that  the  bed  resistance  of  a  loaded 
stream  has  two  parts,  one  due  to  the  texture 
of  the  bed,  the  other  to  the  work  of  traction, 
and  that  the  reduction  of  velocity  by  traction 
could  be  measured  by  comparing  speeds  of 
loaded  and  unloaded  streams  on  beds  of  the 
same  texture  and  slope.  A  series  of  experi- 
ments were  accordingly  performed  with  un- 
loaded streams  flowing  over  beds  composed  of 
fixed  grains  of  debris,  or  d6bris  pavements, 
and  it  was  thought  tliat  their  mean  velocities 
would  be  materially  higher  than  those  observed 
with  streams  otherwise  similar  but  bearing 
loads.  When  the  comparison  was  made,  how- 

1  U.  8.  Geol.  Surrey  Water-Supply  Paper  274,  pp.  102-104, 1911. 

2  The  average  is  on  authority  of  Humphreys  and  Abbot  (Physics  and 
hydraulics  of  the  Mississippi);  the  maximum  is  from  an  observation 
reported  by  3.  A.  Seddon  (Rept.  Chief  Eng.  U.  S.  A.,  1887,  p.  3094)— 
"231.9  grams  per  cubic  foot." 


ever,  it  was  found  that  in  11  out  of  16  experi- 
ments the  observed  velocities  were  higher  witli 
loaded  than'  with  unloaded  streams;  and  the 
average  of  the  16  results  was  of  the  same 
tenor,  ascribing  a  slight  excess  of  velocity  to 
the  loaded  streams.  As  the  estimates  of  mean 
velocity  had  been  based  on  observations  of 
depth,  and  as  the  observations  of  depth  were 
difficult  in  the  case  of  loaded  streams,  it  was 
thought  possible  that  the  result  was  affected 
by  a  systematic  error  in  those  depth  measure- 
ments which  had  been  made  by  means  of  the 
gage  (p.  25).  Examples  were  therefore  sought 
in  which  the  depth  had  been  obtained  by  com- 
paring the  average  height  of  the  water  surface 
during  the  run  with  the  average  height  of  the 
debris  bed  after  the  water  had  been  drawn  off. 
Thirteen  instances  were  found  in  which  such 
better-conditioned  measures  could  be  com- 
pared with  measures  of  unloaded  streams,  and 
in  11  of  the  13  comparisons  the  unloaded 
streams  gave  the  higher  mean  velocities.  The 
average  of  the  13  velocities  found  for  loaded 
streams  was  10  per  cent  less  than  the  corre- 
sponding average  for  unloaded  streams.  The 
best  of  the  available  data,  therefore,  give  evi- 
dence of  the  consumption  of  stream  energy  by 
traction,  but  the  evidence  is  not  so  consistent 
as  to  free  the  matter  from  doubt.  In  Table  80 
Vmi  and  Vmu  represent  the  mean  velocities  of 
loaded  and  unloaded  streams,  and  the  various 
data  are  arranged  according  to  the  magnitude 
of  the  ratio  of  these  velocities.  This  arrange- 
ment brings  out  the  apparent  fact  that  the  re- 
duction of  mean,  velocity  by  traction  is  greater 
for  small  loads  than  for  large,  for  gentle  slopes 
than  for  steep,  for  low  velocities  than  for  high, 
and  for  large  depths  than  for  small.  While  it 
is  entirely  possible  that  such  a  result — a  result 
opposed  to  my  preconceptions — has  been  occa- 
sioned by  a  systematic  error  of  observation,  I 
am  disposed  to  regard  it  rather  as  the  expres- 
sion of  some  physical  law  which  has  escaped 
my  analysis.  It  is  possibly  connected  with  a 
fact  brought  out  in  the  following  chapter  in 
the  discussion  of  vertical  velocity  curves — the 
fact  that  addition  of  load  hjas  a  pronounced 
influence  on  the  distribution  of  velocities,  in- 
creasing the  contrast  between  velocities  near 
tne  bed  and  the  mean  velocity. 

If  the  variations  of  the  load  effect  be  ignored 
and  attention  given  only  to  the  means  of  quan- 
tities compiled  in  the  table,  it  appears  that  10 


230 


TRANSPORTATION    OF   DEBRIS   BY   RUNNING   WATER. 


per  cent  average  reduction  of  velocity  corre-     cubic  foot  of  discharge,  or  0.52  per  cent,  by 
sponds  to  an  average  load  of  148.5  gm./sec.  per     weight,  of  tractional  load. 

TABLE  80. — Comparison  of  mean  velocities  of  streams  with  and  without  tractional  load. 

[Width,  1.00  foot.] 


Grade  of 
debris. 

Dis- 
charge. 

Slope. 

Load. 

Duty. 

Depth  of 
unloaded 
stream. 

Vmi 

Vmu 

Vml 
Vmu 

Ftfjsec. 

Per  cent. 

Gm.lsec. 

Gm.lmc. 

Foot. 

Ft./sec. 

Ft./sec. 

CD) 

0.734 

0.37 

21 

28.6 

0.410 

.79 

2.43 

0.74 

(D) 

.734 

.32 

12 

16.4 

.411 

.79 

2.35 

.76 

(D) 

.363 

.43 

12 

33.1 

.229 

.97 

1.97 

.80 

MS 

.363 

.43 

21 

58.0 

.193 

.88 

2.29 

.82 

(D) 

.363 

.58 

23 

63.4 

.199 

.82 

2.15 

.85 

! 

.363 
.363 

.54 
.35 

25 
12 

69.0 
33.1 

.176 
.224 

2.03 
1.62 

2.39 
1.87 

.85 
.87 

(D) 

.545 

.53 

30 

55.0 

.240 

2.27 

2.46 

.92 

(B) 

.734 

.38 

38 

51.7 

.267 

2.75 

2.92 

.94 

(Bi 

.734 

.48 

52 

70.8 

.256 

2.87 

3.03 

.95 

(B) 

.734 

.54 

80 

109.0 

.241 

3.04 

3.10 

.98 

(D) 

.363 

2.11 

258 

711.0 

.098 

3.70 

3.37 

1.10 

(D) 

.363 

1.94 

229 

632.0 

.099 

3.66 

3.30 

1.11 

148.5 

.90 

THE    TWO    LOADS. 

The  reaction  of  tractional  load  on  velocity 
affects  primarily  and  chiefly  the  zone  of  salta- 
tion, but  there  is  also  a  general  retardation  of 
the  stream.  The  reaction  of  suspended  load 
reduces  all  velocities,  including  those  of  the 
tractional  zone.  Thus  the  magnitude  of  each 
load  affects  capacity  for  the  other  load,  and  it 
also  affects  the  conditions  of  competence  for 
suspension  and  competence  for  traction. 

As  all  parts  of  the  load  influence  velocity,  so 
all  parts  influence  the  general  slope  of  an  allu- 
vial stream,  which  is  automatically  adjusted 
so  as  to  give  to  the  discharge  the  ability  to 
transport  all  the  material,  coarse  and  fine, 
which  is  supplied.  The  adjustment  is  actually 
made  through  the  coarser  material,  for  the  per- 
manent deposits  of  the  stream  bed  are  from 
the  tractional  load. 

Along  with  the  adjustment  of  slope  goes  an 
automatic  partition  of  the  varied  load  into  sus- 
pensional  and  tractional.  If  the  lower  Missis- 
sippi, for  example,  were  to  be  supplied  for  the 
future  with  only  that  part  of  its  load  which  is 
npw  carried  in  suspension,  it  would  so  reduce 
its  slope  that  the  slackened  current  would  drop 
a  portion  of  that  load  and  thereafter  move  it 
by  traction.  If,  on  the  other  hand,  the  river 
were  to  be  deprived  of  the  fine  debris  now  car- 
ried in  suspension,  it  would  so  quicken  its  cur- 
rent as  to  lift  into  suspension  a  portion  of  the 
de'bris  now  carried  by  traction  and  would 
adjust  its  slope  in  such  way  as  to  maintain  the 
partition  of  load. 


An  exception  to  the  general  law  of  automatic 
partition  is  found  when  the  load  has  only  small 
range  in  fineness,  and  this  was  illustrated  by 
the  artificial  conditions  of  the  laboratory;  but 
it  is  not  known  that  natural  streams  illustrate 
the  exceptional  case. 

This  phase  of  river  adjustment  is  well  illus- 
trated by  Yuba  River,  which  I  studied  in 
1904-1908.  Where  it  issues  from  the  moun- 
tains it  carried  a  heavy  load  of  coarse  debris 
with  which  it  was  building  up  its  bed  at  the 
edge  of  the  Sacramento  Valley.  A  dam  thrown 
across  it  in  the  region  of  deposition  arrested, 
the  tractional  load  for  a  time  and  gave  an 
opportunity,  by  the  aid  of  measurements,  to 
estimate  its  amount.  There  were  also  samp- 
lings of  the  water  and  measurements  of  sus- 
pended load.  The  tractional  load  consisted 
mainly  of  gravel,  with  coarse  sand  and  many 
bowlders,  and  the  suspended  load  during  flood 
included  sand  and  finer  debris.  Fifteen  miles 
below,  near  the  mouth  of  the  river,  the  trac- 
tional load  consisted  of  sand,  with  rare  small 
pebbles,  and  only  clay  and  silt  were  in  suspen- 
sion. Here,  too,  the  load  suspended  at  flood 
stage'  was  estimated  from  a  sampling  of  the 
water.  A  computation  based  on  the  various 
data  indicated  that  at  flood  stages  the  sus- 
pended load  was  approximately  equal  to  the 
tractional  load  at  each  of  the  two  localities.1 
While  the  data  for  this  estimate  were  imperfect 
in  many  ways,  they  were  nevertheless  better 
than  any  other  with  which  I  am  acquainted. 

i  A  somewhat  fuller  statement  may  be  found  in  Geol.  Soc.  America 
Bull.,  vol.  18,  pp.  657-658,  1898. 


APPLICATION   TO   NATURAL   STREAMS. 


231 


Information  as  to  suspended  loads  is  fairly 
abundant,  but  there  are  no  satisfactory  data 
as  to  the  complementary  tractional  loads. 

The  character  of  the  partition  of  load  depends 
in  each  particular  instance  on  the  relative 
proportions  of  the  various  grades  in  the  debris 
with  which  the  stream  is  supplied.  It  varies 
from  point  to  point  along  the  course  of  the 
same  stream.  In  the  case  of  Yuba  River  the 
variation  was  duo  to  the  fact  that  exceptional 
conditions  created  by  hydraulic  mining  had 
thrown  the  bed  profile  out  of  adjustment;  and 
many  other  streams  are  dealing  with  new  and 
man-made  conditions;  but  natural  streams 
also  have  strongly  contrasted  load  conditions 
in  different  parts  of  their  courses.  These  arise, 
first,  from  the  diversity  of  detritus  furnished 
by  tributaries,  and,  second,  from  the  gradual 
comminution  of  the  load  as  it  is  borne  along. 

The  partition  of  load  is  greatly  modified  by 
variation  of  discharge  and  by  temporary  con- 
ditions of  debris  supply.  A  number  of  changes 
due  to  varying  discharge  have  already  been 
mentioned.  The  changes  in  velocity  affect  the 
grade  of  fineness  marking  the  plane  of  partition. 
With  reduction  of  discharge  from  a  flood  stage, 
the  conditions  of  partition  come  to  differ  in 
pools  and  over  shoals,  and  usually  suspension 
ceases  altogether,  while  traction  is  still  con- 
tinued on  the  shoals.  With  flood  discharge, 
also,  the  partition  is  probably  not  quite  the 
same  for  deep  and  shoal,  a  portion  of  the  load 
traveling  by  suspension  through  the  deeps  but 
by  traction  over  the  shoals. 

When  a  flood  is  occasioned  by  heavy  rains 
the  fine  particles  of  soil  are  washed  to  the 
stream,  with  the  result  (!)  that  the  suspended 
load  is  relatively  large  and  (2)  that  the  limit- 
ing grade  is  relatively  fine.  When  an  equal 
flood  is  caused  by  snow  melting,  without  rain, 
the  suspended  load  is  smaller  and  the  limiting 
grade  coarser;  in  extreme  cases  there  may  be  no 
suspension  whatever.  While  a  river  is  at  low 
stage  and  without  load,  a  local  shower  may  wash 
to  it  a  temporary  supply  of  debris  of  which  only 
the  finer  part  will  be  immediately  transported, 
giving  suspension  without  traction. 

The  partition  of  load  also  varies  greatly  from 
point  to  point  in  the  same  cross  section,  com- 
petence for  ^suspension  and  traction  both 
responding  to  the  control  of  velocity. 

Despite  these  variable  factors,  it  is  quite 
possible  that  in  the  entire  load  of  a  stream 


largo  enough  to  be  called  a  river  there  is  a 
fairly  definite  ratio  between  suspension  and 
traction..  Knowledge  of  that  ratio  would  be  of 
practical  importance  to  the  engineer  and  geolo- 
gist, because  the  measurement  of  the  tractional 
load  is  always  difficult,  while  measurement  of 
the  suspended  load  is  merely  a  matter  of  rou- 
tine and  patience.  The  estimate  afforded  by 
the  Yuba  is  a  rough  approximation  and  is 
qualified  by  the  fact  that  the  stream  had  not 
an  established  regimen  but  was  engaged  in 
adjusting  its  slope  to  new  conditions  of  load. 

Of  other  estimates  of  tractional  load,  the  one 
most  often  quoted  is  that  of  Humphreys  and 
Abbot.  They  found  that  the  bar  across  one  of 
the  mouths  of  the  Mississippi  increased  its 
width  at  the  top  at  the  rate  of  338  feet  a  year 
and  that  the  material  added  was  similar  to  that 
of  the  tractional  load.  Assuming  that  the 
deposit  had  an  outward  extent  to  the  depth  of 
100  feet  and  a  transverse  extent  equal  to  the 
widths  of  all  the  mouths,  they  made  a  compu- 
tation from  which  the  tractional  load  was 
estimated  at  11  per  cent  of  the  suspended 
load.1 

An  elaborate  study  of  the  loads  of  the  Rhone 
was  made  by  Adolphe  Gu^rard,2  who  measured 
the  suspended  material  by  sampling  the  water 
for  two  years  and  computed  the  total  output 
from  the  soundings  of  the  sea  bed,  the  survey 
of  which  had  been  repeated  after  an  interval 
of  31  years.  He  found  the  suspended  load  less 
than  one-fourth  of  the  whole.  Various  partial 
estimates  have  been  based  on  the  march  of 
subaqueous  dunes.  Where  the  volume  of  a 
dune  and  its  rate  of  progression  are  known, 
their  product  determines  definitely  a  part  of 
the  load,  and  if  the  data  cover  a  year  they 
yield  at  once  a  minimum  estimate  for  the 
annual  load.  Some  large  dunes  in  various 
streams  have  been  reported  to  persist  from 
year  to  year,  with  progressive  change  of  posi- 
tion, but  Arthur  Hider,  who  kept  two  tracts  of 
the  Mississippi  channel  under  observation  for 
about  a  year,  found  that  the  dunes  were  re- 
peatedly readjusted  in  respect  to  size  as  the 
river  stage  changed,  so  that  none  could  be 
identified  through  long  periods,  while  the 
epochs  of  readjustment  were  characterized  by 
general  deposition  or  general  scour,  which 
could  not  be  accounted  for  as  a  result  of  the 

'  Hydraulics  of  the  Mississippi,  p.  149. 

2  Inst.  Civil  Eng.  Proc.,  vol.  82,  pp.  308-310, 1885. 


232 


TRANSPORTATION    OF   DEBRIS  BY   RUNNING    WATER. 


dune  transformations.1  A  similar  remodeling 
was  observed  by  Partiot  in  the  Loire.2  Hider 
estimated  the  whole  movement  in  dunes  as 
14,800  cubic  yards  in  24  hours,  which  corres- 
ponds to  2.2  per  cent  of  the  average  suspended 
load  reported  by  Humphreys  and  Abbot;  but 
his  judgment  was  that  the  entire  movement  of 
debris  along  the  bed  was  at  least  ten  times  as 
great  as  the  movement  in  dunes. 

As  a  generalization  from  extensive  observa- 
tions of  dunes  in  the  river  Loire,  Sainjon  formu- 
lates their  rate  of  advance  when  the  material  is 
sand  as  a  function  of  surface  velocity,  Vs.  In 
his  equation  3 

Advance  =  0.000 13  (  I7,2- 0.11) 

the  units  are  metric.  Substituting  feet  for 
meters,  we  have 

Advance  =  0.00004  (  Fs2-  1.18) 

The  associated  average  height  of  dune  crests  4 
is  0.77  meter,  or  2.54  feet.  As  the  mean  height 
of  a  dune  is  approximately  one-half  the  height 
of  its  crest,  we  may  multiply  the  rate  of  advance 
by  2.54/2  and  obtain 

0.00005  (  T7-1.18) 

as  an  expression  for  the  Loire's  load  of  sand 
carried  in  dunes,  for  each  foot  of  channel  width 
occupied  by.  dunes.  In  a  later  discussion  by 
Lechalas  (see  p.  193)  it  is  shown  that  Sainjon's 
formula  does  not  apply  to  surface  velocities 
above  3.3  ft. /sec.,  the  rate  of  dune  advance  being 
then  checked  because  part  of  the  sand  escapes 
into  the  body  of  the  current  and  is  not  added 
to  the  downstream  faces  of  the  dunes;  but  the 
observations  on  dunes  constitute  the  quanti- 
tative basis  of  the  formulation  of  debris  trans- 
portation by  Lechalas,  of  which  an  account  is 
given  in  Chapter  X. 

In  this  connection  mention  may  be  made  of 
moving  sand  bodies  of  a  different  order  of  mag- 
nitude and  probably  of  a  different  type.  They 
are  greater  than  the  dunes  of  the  same  stream 
and  are  coordinate  in  size  with  the  bars  sepa- 
rating deeps  but  are  distinguished  from  the 

i  Mississippi  River  Comm.  Kept,  for  1882,  pp.  80-88.  Observations 
by  W.  H.  Powless,  made  at  a  different  place  and  in  another  year,  are  o( 
the  same  tenor.  See  idem  for  1881,  pp.  66-120. 

>  Annales  des  ponts  et  chaussees,  M&n.,  Sth  ser.,  vol.  1,  p.  270,  1871. 

a  Quoted,  with  some  of  the  data,  by  Partiot,  idem,  pp.  271-273.  The 
coefficient  is  there  erroneously  given  as  0.0013. 

<  Given  by  Lechalas  in  the  same  volume,  pp.  387-388. 


bars  by  their  migration  downstream.  In  some 
of  the  "regularized  "  streams  of  Europe-,  where 
the  main  channel  is  artificially  restricted  to 
curves  of  large  radius,  they  are  developed  in 
systematic  alternation  at  the  two  sides,  and  the 
thalweg  winds  between  them.5  In  the  Missis- 
sippi they  sometimes  appear  in  the  reaches. 
Their  progress  downstream  is  accomplished  by 
deposition  on  forward  slopes  and  erosion  of  rear 
slopes,  but  the  forward  slopes  are  not  steep,  like 
those  of  dunes,  and  their  material  is  not  wholly 
deri ved  from  the  traction al  load.  Blasius  8  re- 
gards them  as  essentially  dunes,  correlating 
them  specifically  with  dunes  of  reticulated  pat- 
tern. My  own  view,  not  necessarily  inconsist- 
ent with  his,  connects  them  with  the  fixed  bars 
separating  the  deeps  of  a  meandering  stream. 
A  free  stream  does  not  tolerate  a  straight  chan- 
nel. If  a  straight  channel  of  moderate  width  be 
given  to  a  stream,  the  current  swings  rhyth- 
mically to  right  and  left,  and  if  the  banks 
yield  it  develops  meanders.  The  meanders 
then  migrate,  according  to  laws  of  their  own, 
and  the  bars  are  fixed  in  relation  to  the  mean- 
ders. If  the  banks  do  not  yield,  the  system  of 
shoals  and  deeps  established  by  the  swinging 
current  migrates  slowly  downstream.  It  is 
evident  that  the  migration  of  these  shoals  is 
one  of  the  factors — and  may  be  an  important 
factor — in  the  work  of  transportation ;  and  also 
that  every  measurement  of  the  migration  of  a 
shoal  is  a  partial  measurement  of  load. 

Pilots  of  Mississippi  steamboats  observe 
that  the  bars  at  crossings  are  built  up  by  floods, 
and  such  changes  have  been  measured  by 
engineers.  The  generalization  has  sometimes 
been  made  that  deposition  is  a  specific  function 
of  floods,  but  a  more  satisfactory  interpretation 
is  given  by  McMath,7  who  maintains  that  the 
rising  river  scours  from  the  deeps  to  deposit  on 
the  shoals,  and  the  falling  river  scours  from 
the  shoals  to  deposit  in  the  deeps.  The 
transfers  are  the  joint  work  of  traction  and 
suspension.  As  such  changes  of  the  stream 
bed  are  measurable  they  afford  quantitative 
data  as  to  load,  and  it  was  from  their  observa- 
tion that  Hider,  as  previously  quoted,  inferred 
that  the  dune  movement  in  the  Mississippi 
includes  but  a  small  fraction  of  the  tractional 
load. 

t  Engels,  H.,  Zeitschr.  Bauwesen,  vol.  55,  pp.  604-680, 1905. 

s  Idem,  vol.  CO,  pp.  4(3-472, 1910. 

i  Mississippi  River  Comm.  Kept,  for  18X1,  p.  252. 


APPLICATION    TO   NATURAL   STREAMS. 


233 


The  dune  movement,  the  migrations  of 
greater  bars,  and  the  transfers  of  debris  from 
deep  to  shoal  and  shoal  to  deep  are  all  compe- 
tent to  give  information  as  to  tractional  load, 
but  the  estimates  they  give  are  minimum 
estimates,  to  be  supplemented  by  estimates  of 
the  material  which  at  flood  stages  is  swept 
steadily  along  without  contributing  to  any  of 
the  temporary  deposits  in  such  way  as  to  be 
accessible  to  measurement. 

AVAILABILITY  OP  LABORATORY  RESULTS. 

THE    SLOPE    FACTOR. 

Wo  are  now  ready  to  inquire  whether,  in 
view  of  the  diversities  and  complexities  affect- 
ing traction  by  natural  streams,  the  formula 
for  tractional  capacity  derived  under  the 
comparatively  simple  conditions  of  the  labora- 
tory is  of  practical  value  in  connection  with 
natural  streams.  The  four  factors  of  the 
formula  may  first  be  considered  separately. 

The  general  slope  of  a  stream  is  the  quotient 
of  fall  by  distance,  the  distance  being  taken 
along  the  stream's  course.  It  is  best  measured 
at  high  stage,  because  the  chief  work  of  grading 
the  channel  is  accomplished  by  floods. 

With  reference  to  variations  in  capacity  at  a 
single  locality,  slope  does  not  enter,  the  varia- 
tions being  referred  to  discharge;  but  account 
must  be  taken  of  slope  in  comparing  different 
divisions  of  the  same  stream  and  in  comparing 
one  stream  with  another. 

In  all  such  cases  the  stream's  slope  is  as 
definite  a  quantity  and  is  susceptible  of  as 
precise  measurement  as  is  the  slope  of  the 
laboratory  channel.  It  differs  as  to  its  repre- 
sentative character.  The  laboratory  slope  is 
connected  with  a  single  discharge  and  a  single 
grade  of  debris  of  determinable  fineness.  The 
slope  of  the  natural  stream  does  not  represent 
the  adjusting  work  of  a  determinable  discharge 
but  is  a  compromise  product  of  the  work  of 
many  discharges,  and  it  is  usually  true  that 
the  velocities  associated  with  these  discharges 
have  determined  equally  diverse  mean  fine- 
nesses of  debris.  The  work  of  the  natural 
stream,  moreover,  has  been  characterized  by 
greater  diversity,  from  point  to  point,  of  the 
bed  velocities,  and  its  system  of  velocities  has 
been  regulated  in  part  by  suspended  load. 

These  difficulties  would  prove  insuperable  if 
attempt  were  made  to  infer  the  capacity  of  a 
natural  stream  from  that  of  a  laboratory 


stream,  but  they  are  not  necessarily  important 
in  transferring  a  law  of  variation  from  a  group 
of  laboratory  streams  to  an  equally  harmonious 
group  of  naturalstreams.  If  the  diversification 
of  discharges  and  finenesses  is  of  the  same  type 
for  the  examples  of  natural  streams  between 
which  comparison  is  made,  it  may  well  be  that 
the  slopes  are  comparable,  one  with  another, 
in  the  same  sense  in  which  they  are  comparable 
in  laboratory  work,  and  that  their  relations  to 
capacity  should  follow  the  same  law. 

THE    DISCHARGE    FACTOR. 

Discharge  differences  must  be  considered 
when  the  tractional  capacities  of  different 
streams  are  to  be  compared,  and  also  in  com- 
paring the  capacities  of  the  same  stream  at 
different  tunes. 

In  making  comparison  between  different 
streams  it  is  important  that  the  discharges  used 
be  coordinate — that  is,  that  they  represent 
equivalent  phases  of  stream  work.  If  co- 
ordination be  not  secured,  allowance  must  be 
made  in  one  stream  or  the  other  for  the  varia- 
tion of  capacity  with  stage.  In  case  the  prob- 
lem is  such  that  the  choice  of  phase  is  optional, 
preference  should  be  given  to  flood  phases, 
because  the  general  slope  and  the  details  of 
channel  shape  are  approximately  adjusted  to 
such  phases.  The  greatest  known  discharge  is 
probably  less  representative  of  the  channel 
conditions  than  is  the  mean  of  annual  maxima 
of  discharge.  It  is  believed  that  with  use  of 
discharges  that  are  both  representative  and  co- 
ordinate the  discharge  factor  of  the  empiric 
formula  may  be  applied.  The  result  of  such 
application  will  be  the  more  satisfactory  in 
proportion  as  the  streams  compared  are  allied 
in  type  and  will  be  relatively  unsatisfactory  for 
streams  in  different  climatic  provinces  or  for 
comparison  of  a  direct  alluvial  stream  with 
one  which  meanders. 

It  is  to  be  observed  that  in  ah1  studies  of  allu- 
vial streams  the  discharge  of  which  account 
should  be  taken  in  connection  with  traction  is 
the  discharge  flowing  in  the  channel  proper. 
That  which  passes  the  banks  ceases  to  con- 
tribute of  its  energy  to  the  work  of  traction, 
and  the  portion  of  load  diverted  with  it  is  not 
tractional. 

The  case  of  variation  of  discharge  in  the 
same  stream  is  complicated  by  simultaneous 
variations  of  fineness  and  competence.  In  the 


234 


TRANSPORTATION    OF   DEBRIS   BY   RUNNING   WATER. 


experiments  with  sieve-separated  grades  of 
debris  fineness  and  competence  were  constants 
with  reference  to  discharge;  but  in  a  natural 
stream,  where  the  tractional  load  may  have 
great  range  in  fineness,  the  mean  fineness  of  the 
load  varies  with  discharge,  and  the  reason  of 
its  variation  is  that  competence  varies  with 
discharge.  The  two  competences  which  limit 
the  range  in  fineness  move  up  and  down  as 
discharge  changes,  and  the  mean  fineness 
moves  with  them.  Therefore  the  response  of 
capacity  to  discharge  can  not  be  considered  by 
itself. 

For  convenience  in  analyzing  the  conditions, 
let  us  assume  first  a  discharge  which  is  adjusted 
to  the  details  of  channel  form,  to  the  deeps  and 
shoals.  If,  now,  the  discharge  be  increased,  and 
with  it  the  whole  system  of  velocities,  trans- 
portation will  be  everywhere  stimulated,  part 
of  the  tractional  load  will  join  the  suspended, 
and  the  scouring  of  the  deeps  will  bring  into 
the  tractional  load  a  greater  proportion  of  the 
coarser  elements  of  the  load.  The  mean  fine- 
ness of  that  load  will  be  reduced,  and  the  ca- 
pacity, while  enlarged  by  increase  of  discharge, 
will  be  somewhat  reduced  by  loss  of  fineness. 
The  increase  with  discharge  will  be  less  than  if 
the  fineness  were  constant. 

If,  on  the  other  hand,  the  discharge  be  re- 
duced, some  of  the  coarser  material  comes  to 
rest,  while  finer  debris  is  added  from  the 
suspended  load.  So  the  reduction  of  capacity 
from  diminished  discharge  is  qualified  by  the 
effect  of  increased  fineness.  But  before  the 
change  in  discharge  has  gone  far  the  deeps 
become  pockets  for  the  reception  of  deposits, 
and  traction  is  restricted  to  the  intervening 
shoals,  where  it  causes  erosion.  The  erosion 
is  selective,  leaving  an  ever-increasing  assem- 
blage of  residuary  coarse  material,  which  tends 
to  protect  the  finer.  The  current  on  the  shoals 
no  longer  obtains  a  full  supply  of  the  material 
for  which  it  is  competent,  and  the  load  and 
capacity  part  company.  Or  we  may  say  that 
as  the  erosion  of  the  shoals  progresses  the  mean 
fineness  of  the  accessible  debris  is  reduced  until 
a  grade  is  reached  for  which  the  current  is  not 
competent.  In  either  case  the  decadence  of 
traction  follows  a  law  which  is  not  well  repre- 
sented by  the  discharge  term  of  the  laboratory 
formula.  . 

The  above  analysis  postulates  a  wide  range 
and  somewhat  equable  distribution  of  fineness 


in  the  debris  of  the  stream  bed,  a  condition  not 
always  found.  It  might  not  apply,  for  ex- 
ample, to  a  stream  which  drains  a  district  of 
friable  sandstone  and  is  therefore  supplied  with 
nothing  coarser  than  sand.  Nor  would  it  apply 
well  to  a  stream  supplied  with  very  coarse  and 
very  fine  debris  but  not  well  supplied  with 
intermediate  grades. 

In  most  alluvial  streams,  and  probably  in  all 
meandering  streams,  the  work  of  traction  which 
is  accomplished  on  the  shoals  at  low  stage  and 
midstage  is  almost  negligible  in  comparison 
with  the  high-stage  traction.  Not  only  is  the 
rate  of  traction  slow,  but  the  field  of  traction 
is  restricted.  If  a  single  formula  will  not  fit 
both  low  and  high  stages,  the  one  adjusted  to 
high-stage  variations  will  have  the  greater 
practical  value. 

Yet  another  consideration  enters  here,  and 
one  of  peculiar  importance.  When  discharge  is 
reduced,  and  the  competence  of  the  current  for 
traction  is  thereby  changed,  the  coarse  material 
eliminated  from  the  tractional  range  ceases  to 
be  transported;  but  when  discharge  is  increased, 
and  the  competence  of  the  current  for  suspen- 
sion is  thereby  changed,  the  fine  material 
eliminated  from  the  tractional  range  continues 
to  be  transported.  It  is,  in  fact,  transported 
more  rapidly,  so  that  a  greater  amount  passes 
a  given  section  each  second.  For  most  or  all 
practical  purposes  the  change  in  mode  of  trans- 
portation is  of  no  moment,  and  those  purposes 
would  be  served  by  a  formula  which  should 
include  the  material  shifted  and  ignore  the 
change  in  mode.  In  the  system  of  reactions 
set  up  by  change  of  discharge  the  two  modes 
of  transportation  are  so  interwoven,  in  fact, 
that  the  practical  discrimination  of  the  sus- 
pended and  tractional  loads  is  impossible. 
Even  in  the  laboratory  experiments  devised 
specially  for  the  study  of  traction  a  certain 
amount  of  interplay  was  tolerated,  for  tem- 
porary suspension  appeared  over  the  crests  of 
some  of  the  antidunes  and  also  in  the  bends  of 
the  crooked  channels. 

If  the  purely  tractional  point  of  view  is  to 
be  exchanged  for  another,  what  shall  be  sub- 
stituted ?  One  natural  suggestion  is  to  include- 
in  a  single  view  the  entire  load,  suspended  and 
tractional;  another  to  include  along  with  the 
tractional  only  that  part  of  the  suspended  load 
which  for  part  of  the  time  is  tractional  also. 
That  which  would  be  included  in  one  view  and 


APPLICATION    TO   NATURAL   STREAMS. 


235 


excluded  from  the  other  is  the  finer  part  of  the 
suspended  load,  the  part  that  does  not  sink  to 
the  bottom  so  long  as  the  current  is  sufficiently 
active  for  traction.  Being  purely  suspensional, 
its  quantity  is  peculiarly  a  function  of  supply 
and  is  connected  with  discharge  only  through 
the  association  of  discharge  with  rain.  Wher- 
ever discharge  is  largely  a  matter  of  tribute 
from  snowbanks,  the  suspended  load  is  con- 
spicuously independent  of  discharge. 

If  we  exclude  from  view  the  purely  suspen- 
sional  material,  a  natural  criterion  for  inclusion 
is  the  finest  debris  which  low-stage  discharge 
moves  by  traction,  and  as  low-stage  traction  is 
limited  to  the  bars,  or  interpool  shoals,  it  is 
the  finest  tractional  debris  of  those  shoals.  If 
we  consider  a  gradual  increase  of  discharge  from 
least  to  greatest,  we  have  at  first  no  traction. 
Then  for  a  particular  discharge,  which  may  be 
called  the  competent  discharge,  traction  begins 
on  the  shoals,  only  the  finest  of  the  de'bris  being 
moved.  Gradually  coarser  and  coarser  mate- 
rial is  included,  the  range  in  fineness  and  the 
load  increasing  together;  but  in  this  phase  of 
action  the  load  is  not  necessarily  the  equiv- 
alent of  the  capacity,  for  it  may  be  limited  by 
the  supply  of  de'bris  of  requisite  fineness. 
After  a  time  another  critical  discharge  is  at- 
tained, which  initiates  the  loaning  of  de'bris 
from  traction  to  suspension,  and  thereafter  a 
constantly  increasing  share  of  the  traveling 
de'bris  is  suspended.  As  the  suspended  parti- 
cles travel  faster  than  the  saltatory,  and  as 
capacity  is  the  ability  of  the  stream,  measured 
in  grams  per  second,  to  move  de'bris  past  a 
sectional  plane,  the  transfer  from  traction  to 
suspension  is  an  important  factor  in  the  en- 
hancement of  capacity. 

The  relation  of  capacity  to  discharge,  con- 
templated from  this  viewpoint,  has  two  ele- 
ments in  common  with  the  discharge  factor  of 
the  laboratory  formula.  It  includes  a  compe- 
tent discharge,  corresponding  to  the  zero  of 
capacity,  and  it  associates  continuous  increase 
of  capacity  with  continuous  increase  of  dis- 
charge. It  differs,  however,  in  important  ways, 
and  the  possibility  of  expressing  it  by  a  definite 
formula  is  not  evident.  In  the  pool  and  rapid 
phase  of  activity  the  supply  of  de'bris  suitable 
for  traction  is  usually  limited,  and  in  many 
streams  it  is  exhausted  during  each  recurrence 
of  the  phase.  In  the  phases  of  greater  dis- 
charge, when  traction  occurs  in  the  deeps  as 


well  as  on  the  shoals,  the  sequence  of  capacities 
depends  not  only  on  discharge  but  on  the  rela- 
tive proportions  of  debris  of  different  grades  of 
fineness  in  the  material  of  the  load.  It  is  prob- 
able that  for  most  streams  the  load-discharge 
function  is  discontinuous  at  the  limit  of  the 
pool  and  rapid  phase. 

Because  of  this  presumable  discontinuity, 
because  the  tractional  work  while  the  pools 
exist  accomplishes  only  a  local  transfer  of 
de'bris,  and  because  the  work  performed  is 
usually  of  negligible  amount  in  comparison 
with  the  work  of  larger  discharges,  it  is  prob- 
ably better  to  ignore  altogether  the  pool  and 
rapid  phase  in  any  attempt  at  general  formu- 
lation. If  that  be  left  out  of  account  and  if 
the  general  features  of  the  laboratory  formula 
be  retained,  the  constant  «  becomes  the  dis- 
charge which  initiates  traction  in  the  deeps, 
and  thus  initiates  through  transportation  of 
bottom  load.  If  we  accept  that  as  a  starting 
point,  the  material  so  fine  as  to  be  suspended 
by  that  discharge  may  be  classed  as  purely  sus- 
pensional,  and  other  material  suspended  by 
larger  discharges  may  be  grouped  with  the 
tractional  load.  For  the  tractional  load  thus 
enhanced,  or  the  amplitractional  load,  as  it 
may  conveniently  be  called,  the  rate  of  varia- 
tion with  discharge  is  evidently  higher  than 
the  rates  found  for  simple  grades  in  the  labora- 
tory, and  it  may  be  much  higher,  for  the  de'bris 
diverted  from  traction  to  suspension,  instead 
of  lagging  behind  the  lowest  and  slowest 
threads  of  the  current,  now  speeds  with  the 
current's  mean  velocity. 

It  is  possible  that  a  practical  formula  for  the 
fluctuations  of  an  alluvial  river's  load  may  fol- 
low these  lines,  taking  the  form 


where  Ca  is  the  capacity  for  amplitractional 
load,  and  K,  is  the  smallest  discharge  competent 
to  establish  a  continuous  train  of  traction 
through  deeps  and  shoals;  but  the  suggestion 
as  to  form  has  no  better  basis  than  analogy, 
and  no  data  are  known  tending  to  determine 
the  magnitude  of  the  important  parameter  o. 

THE    FINENESS    FACTOR. 

When  the  work  of  two  natural  streams  is 
compared  and  the  streams  are  of  the  same  type, 
it  i-i  believed  that  the  fineness  factor  of  the 


236 


TRANSPORTATION    OF   DEBRIS  BY   RUNNING    WATER. 


laboratory  formula  is  applicable.  It  is  true 
that  fineness  enters  in  a  relatively  complex 
way  into  the  determination  of  the  loads  of 
natural  streams,  but  for  the  comparison  indi- 
cated the  elements  of  influence  are  severally 
represented  by  the  experiments,  and  their  to- 
tals should  follow  a  law  of  the  same  type.  For 
small  discharges  this  inference  is  subject  to 
certain  qualifications,  which  will  appear  from 
what  follows. 

When  the  work  of  the  same  stream  is  com- 
pared under  different  discharges,  a  difference  in 
fineness  is  developed  under  the  laws  of  compe- 
tence. With  larger  discharge  the  mean  fine- 
ness is  less  than  with  small  discharge,  and  the 
difference  in  fineness  conspires  with  the  differ- 
ence in  discharge  to  determine  capacity.  For 
reasons  explained  in  the  last  section,  however, 
capacity  can  not  always  be  considered  synony- 
mous with  load  when  the  discharge  is  small. 

THE    FORM-RATIO    FACTOR. 

In  the  reaches  of  a  direct  alluvial  stream  there 
is  approximate  uniformity  of  depth  at  high 
stage,  and  the  conditions  involving  form  ratio 
are  essentially  like  those  realized  in  the  labora- 
tory. To  such  cases  the  principles  developed 
in  the  laboratory  studies  should  be  applicable. 

It  is  true  in  a  general  way,  as  already  men- 
tioned on  page  223,  that  at  a  high  stage  of  a 
natural  stream  the  sectional  area  is  about  the 
same  for  the  reaches  as  for  the  bends,  and  so 
too  is  the  width.  It  follows  that  the  mean 
depth  is  about  the  same,  although  the  maxi- 
mum depth  may  be  very  different.  The  high- 
stage  capacity  is  also  the  same  at  every  sec- 
tion, after  the  channel  form  has  been  adjusted 
to  the  discharge.  If  these  generalizations  are 
correct,  the  principle  involved  in  the  form-ratio 
factor  of  the  laboratory  formula  is  applicable  to 
curving  streams,  provided  form  ratio  is  inter- 
preted as  the  ratio  of  mean  depth  to  width,  and 
not  as  the  ratio  of  maximum  depth  to  width. 

In  the  analysis  of  conditions  determining  the 
relation  of  capacity  to  form  ratio  (Chapter  IV) 
an  important  role  was  ascribed  to  the  resistance 
of  the  banks;  and  the  quantity  of  that  resist- 
ance was  represented  in  one  of  the  parameters 
of  the  formula,  ot.  The  optimum  form  ratio, 
p,  was  found  to  vary  inversely  with  at  and,  there- 
fore, to  vary  inversely  with  the  resistance  of  the 
banks.  The  resistance  afforded  by  river  banks 


is  greater  than  that  given  by  the  smooth  walls 
of  laboratory  channels,  and  this  element  tends 
to  make  the  optimum  form  ratio  relatively 
small  for  rivers.  Its  influence,  however,  is  over- 
shadowed by  those  of  slope  and  discharge.  As 
the  optimim  ratio  varies  inversely  with  slope, 
and  as  most  rivers  have  lower  slopes  than  the 
experimental  streams,  the  general  tendency  of 
the  slope  element  is  to  make  the  ratio  large 
for  rivers.  As  the  optimum  ratio  varies  in- 
versely with  discharge,  and  as  the  discharges  of 
natural  streams  are  relatively  large,  the  tend- 
ency of  this  element  is  to  make  the  ratio  small 
for  natural  streams.  The  rates  of  variation 
being  unknown,  the  net  result  of  the  three  in- 
fluences can  not  be  inferred  deductively.  The 
data  from  Yuba  River,  cited  in  Chapter  IV  (p. 
135),  show  that  for  one  case  of  a  natural  stream 
the  optimum  ratio  is  decidedly  larger  than  that 
established  by  the  stream  in  its  alluvial  phase 
and  is  of  the  order  of  magnitude  of  the  determi- 
nations made  in  the  laboratory. 

THE  FOUR  FACTORS  COLLECTIVELY. 

The  results  of  the  preceding  discussions  ad- 
mit of  a  certain  amount  of  generalization. 
When  different  streams  of  the  same  type  are 
compared,  and  especially  when  the  type  is  al- 
luvial, the  law  of  their  relative  capacities  at 
high  stage  may  be  expressed  by  the  laboratory 
formula  (109).  The  ability  of  that  formula  to 
express  the  variation  of  capacity  with  discharge 
in  the  same  stream  is  problematic. 

It  has  not  been  shown  that  the  system  of 
numerical  parameters  determined  for  laboratory 
conditions  can  be  used  in  extending  the  appli- 
cation of  the  formula  to  natural  streams.  If 
the  formula  were  rational,  the  result  of  an  ade- 
quate mathematical  treatment  of  the  physical 
principles  involved,  the  constants  measured  in 
the  laboratory  would  be  of  universal  application 
(with  moderate  qualification  for  the  conditions 
imposed  by  the  curvature  of  natural  channels) ; 
but  the  constants  of  an  empiric  formula  afford 
no  basis  for  extensive  extrapolation. 

THE    HYPOTHESIS    OF    SIMILAR    STREAMS. 

When  the  Berkeley  experiments  were  planned 
it  was  assumed  that  the  relations  of  capacity  to 
various  conditions  would  be  found  to  be  sim- 
ple, and  that  the  laboratory  streams  were  rep- 
resentative of  natural  streams  except  as  to  tie 


APPLICATION    TO   NATURAL  STREAMS. 


237 


characters  associated  with  bending  channels. 
Because  of  the  discovered  complexity  of  the 
l:;ws  affecting  capacity  it  is  now  apparent  that 
the  laboratory  formulas  can  not  be  applied  to 
streams  in  general.  It  is,  however,  probable 
that  among  the  great  variety  of  natural  streams 
there  is  a  more  or  less  restricted  group  which 
is  in  such  respect  similar  to  the  laboratory  group 
that  the  empiric  results  of  the  laboratory — or 
at  least  the  results  embodied  in  exponents — 
may  properly  be  applied  to  it . 

The  criteria  of  similarity  between  large  and 
small  have  been  discussed  to  some  extent  by 
others  in  connection  with  the  investigation  of 
hydraulic  problems  by  means  of  models. 
William  Froude  inferred  from  theoretic  con- 
siderations that  if  the  speed  of  a  ship  and  the 
speed  of  its  miniature  model  "  are  proportional 
to  the  square  roots  of  the  dimensions,  their  re- 
sistances at  those  speeds  will  be  as  the  cubes 
of  their  dimensions,"1  and  he  afterward  veri- 
fied this  result  by  experiments.  T.  A.  Hear- 
son,  in  projecting  a  model  river  for  the  inves- 
tigation of  various  hydraulic  problems,  dis- 
cussed separately  the  resistance  to  flow  by  the 
wetted  perimeter,  the  influence  of  varying  sec- 
tional area,  and  the  influence  of  bends.  He 
concluded  that  if  the  linear  dimensions  were 
kept  in  the  same  proportion,  so  that  the  river 
channel  and  its  model  were  similar  in  the  geo- 
metric sense,  the  velocities  would  be  related  f.s 
the  square  roots  of  the  linear  dimensions,  and 
the  discharges  as  the  2.5  powers  of  the  linear  di- 
mensions. It  would  be  necessarythat  thorough- 
nesses of  the  channel  surfaces  have  the  same  dif- 
ferences as  the  linear  dimensions,  and  that  the 
movable  debris  of  the  bed  r.lso  follow  the  laws 
of  linear  dimensions.2  His  deductions  were  not 
tested  by  the  construction  and  use  of  a  model, 
but  they  derive  a  large  measure  of  support  from 
the  verification  of  Froude's  analogous  theorem. 
So  far  as  I  am  aware,  r.ll  the  models  actually 
constructed  to  represent  rivers  and  tidal  basins 
have  been  given  an  exaggerated  vertical  scale.3 
O.  Reynolds  4  made  a  series  of  models  of  tidal 
basins  in  which  the  scales  of  depth  and  of  tidal 

1  These  words  are  quoted  from  Inst.  Naval  Arch.  Trans.,  vol.  15,  p.  151, 
1874.    I  have  not  seen  Froude's  original  discussion  of  the  subject. 

2  Inst.  Civil  Eng.  I'roc.,  vol.  146,  pp.  21G-222,  1900-1901. 

•  See  Fargue,  L.,  La  forme  du  lit  des  rivieres  a  fond  mobile,  pp.  57, 128, 
1908.    Fargue  recommended  for  a  model  river  a  vertical  scale  of  1:100 
and  a  horizontal  scale  of  1:20,  from  which  he  deduced  a  discharge  ratio 
of  1:3,200  and  a  velocity  ratio  of  1:16. 

*  British  Assoc.  Adv.  Sci.  Repts.  1887,  pp.  555-502;  1889,  pp.  328-343; 
I860.  pp.  512-534;  1891,  pp.  386-404. 


amplitude  were  greater  than  the  scale  of  length, 
the  ratios  ranging  from  31:1  to  105:1.  No  ad- 
justment was  made  as  to  size  of  debris,  the  re- 
quirements of  his  investigation  being  met  by 
any  material  fine  enough  to  be  moved  by  the 
currents.  A  tidal  oscillation  was  communi- 
cated to  water  resting  on  a  level  bed  of  sand, 
with  the  result  that  the  bed  was  gradually 
molded  into  shapes  more  or  less  characteristic 
of  estuaries.  From  general  considerations  a 
"law  of  kinetic  similarity"  was  deduced: 


=  constant 


Lt 


in  which  p  is  the  tidal  period,  h  the  depth  of 
water  (proportional  to  the  amplitude  of  the 
tide),  and  L  the  length  of  the  estuary.  Under 
this  law  the  results  were  generally  consistent, 
but  there  was  found  to  be  a  limit  to  the  range 
of  suitable  conditions,  and  this  limit  was  formu- 
lated by 

Jt3e  =  constant 

in  which  e  is  the  exaggeration  of  the  vertical 
scale. 

Eger,  Dix,  and  Seifert,5  making  a  model  of 
a  portion  of  Weser  River  for  the  purpose  of 
studying  the  effect  of  projected  improvements, 
adopted  1 : 100  as  the  scale  of  horizontal  dimen- 
sions and  depths,  and  1 : 6.7  as  the  scale  of  mean 
diameters  of  debris  particles  composing  the 
channel  bed.  It  was  then  a  matter,  first  of 
theory  and  computation  but  finally  of  trial,  to 
select  scales  for  discharge  and  slope.  The  main 
condition  to  be  satisfied  was  that  for  discharges 
corresponding  to  high  and  low  stages  the  depths 
of  water  should  be  properly  related,  according 
to  the  scale  of  linear  dimensions.  For  the  scale 
of  discharges  1 : 40,000  was  finally  adopted,  and 
for  slopes  650 : 1 .  The  resulting  ratio  of  veloci- 
ties was  1:4;  and  this  ratio,  combined  with  the 
ratio  of  debris  sizes,  was  found  to  give  a  time 
ratio  (for  the  accomplishment  of  similar 
changes  in  the  bed  of  the  stream)  of  1:360. 
The  scale  of  velocities  being  only  1:4  while  the 
scale  of  distances  was  1:100,  there  was  an 
exaggeration  of  velocities  in  the  ratio  of  25: 1.6 

The  quantities  of  debris  moved  being  in  the 
ratio  of  1:100 3,  the  distances  moved  in  the 

«  Zeitschr.  Bauwesen,  vol.  56,  pp.  323-344, 1906. 

•  So  stated  by  the  authors.  An  allowance  for  the  general  principle 
that  velocities  are  proportional  to  the  square  root  of  the  hydraulic  mean 
depth,  and  therefore  to  the  square  root  of  linear  dimensions,  would 
indicate  1:10  as  the  normal  ratio  of  velocities,  and  give  2.5:1  as  the 
exaggeration. 


238 


TRANSPORTATION   OF   DEBKIS  BY   RUNNING   WATER. 


ratio  of  1 : 100,  and  the  times  consumed  in  the 
ratio  of  1:360,  the  ratio  of  loads  (per  second) 

wasl:Q™      or   1:280,000.     The  results   were 
ooU 

satisfactory;  it  was  found  that  the  successive 
forms  given  to  the  river  bed  by  variations  of 
discharge  were  repeated  in  the  model. 

The  exaggerations  of  the  vertical  scale  by 
Reynolds  and  of  the  slope  by  Eger,  Dix,  and 
Seifert  had  the  important  effect  of  shortening 
the  time  necessary  to  produce  the  desired  re- 
modeling of  the  mobile  bed.  The  absolute 
proportionality  adopted  by  Froude  and  recom- 
mended by  Hearson  would  have  entailed  a 
prohibitive  consumption  of  time  and  might 
have  added  a  serious  complication  in  connec- 
tion with  the  use  of  very  fine  debris.1 

The  similarity  controlled  by  Reynold's  law 
was  a  relation  between  the  wave  periods  and 
dimensions  of  tidal  basins  and  is  not  closely 
related  to  similarity  in  the  control  of  trac- 
tional  load.  The  similarities  obtained  in  con- 
structing the  model  of  the  Weser  are  more  in 
point,  because  they  involve  an  average  rate  of 
movement  of  debris;  but  they  throw  no  light 
on  the  laws  of  variation  of  debris  movement* 
which  is  the  important  matter  in  bridging  the 
interval  between  our  experiment  streams  and 
natural  streams.  After  attempting  to  use 
various  suggestions  which  came  from  the 
adjustments  of  the  Weser  model,  I  have  re- 
turned to  the  principles  of  geometric  similarity 
employed  by  Froude  and  Hearson. 

Let  us  assume,  as  possible  or  plausible,  that 
the  principles  developed  in  the  laboratory, 
together  with  all  parameters  which  are  of  the 
nature  of  ratios,  are  independent  of  the  scale 
of  operations  and  may  be  applied  to  streams  of 
far  greater  magnitude,  provided  all  linear  fac- 
tors are  magnified  in  equal  degree.  If  width 
and  depth  are  enlarged  in  the  same  ratio,  the 
form  ratio  is  unchanged.  If  longitudinal  dis- 
tance and  loss  of  head  are  enlarged  in  the  same 
ratio,  the  slope  is  unchanged.  If  the  dimen- 
sions of  the  transported  particles  are  enlarged 
in  the  same  ratio  as  the  linear  elements  of  chan- 
nel, the  linear  coarseness  is  increased,  or  the 
linear  fineness  is  reduced  in  that  ratio. 

The  natural  streams  which  may  be  consid- 
ered as  similar  to  the  experimental  streams 
constitute  a  class  of  moderate  size.  The  form 

'  Such  considerations  as  these  affected  the  selection  of  materials  for  the 
Berkeley  experiments  land  prevented  the  employment  of  very  gentle 
slopes. 


ratio  for  rivers  ranges  lower  than  for  experi- 
mental streams,  but  there  is  some  overlap. 
The  smaller  of  the  form  ratios  of  the  laboratory 
are  representative  of  a  considerable  number  of 
rivers.  The  slopes  of  rivers  range  lower  than 
for  laboratory  streams,  but  here  again  there  is 
overlap,  and  the  natural  streams  which  are 
similar  in  respect  to  slope  are  in  general  such 
as  have  coarse  debris,  so  that  there  may  be 
correspondence  in  that  regard  also.  The  simi- 
lar natural  streams  to  which  hypothesis  extends 
the  laboratory  results  are  those  of  large  form 
ratio  and  steep  slope,  carrying  coarse  debris. 

The  primary  difference  between  a  large 
stream  and  a  small  one  being  one  of  discharge, 
and  our  general  inquiry  being  directed  to  the 
valuation  of  capacity  for  traction,  let  us  seek 
an  expression  for  the  relation  of  capacity  to 
discharge  when  similar  streams  of  different  size 
are  compared. 

The  laboratory  data  determine  control  of 
capacity  by  slope,  discharge,  fineness,  and  form 
ratio.  In  similar  streams  slope  and  form  ratio 
are  constant,  and  we  need  consider  here  only 
discharge  and  fineness.  As  we  are  comparing 
the  laboratory  streams  as  a  group  with  similar 
natural  streams,  also  taken  as  a  group,  it  is 
advisable  to  employ  a  mode  of  formulation 
which  lends  itself  to  the  use  of  averages,  and 
the  most  convenient  is  that  of  the  synthetic 
index.  In 


lia  and  7to  are  average  values  of  the  synthetic 
index  and  may  be  estimated,  from  data  in 
Tables  32  and  43,  as  1.32  and  0.77.  Designat- 
ing elements  of  the  larger  and  smaller  streams 
severally  by  subscripts  „  and  ,  ,  we  have,  from 
the  above, 


(113) 


If  we  designate  by  L  the  ratio  between  a  linear 
dimension  of  the  larger  stream  and  the  corre- 
sponding dimension  of  the  smaller. 


d, 


L 


Calling  mean  velocity  V,  bearing  in  mind  that 
the  hydraulic  mean  radius  is  a  linear  dimension 
of  channel,  and  recalling  that  the  Chezy  formula 


APPLICATION    TO    NATUKAL   STREAMS. 


239 


makes   V  vary  as  the  square  root  of  the  hy-      periments  treat  with  confidence  is  from  0.5  to 

draulic  radius,  we  have  3.0  per  cent.     Direct  appli cation  is  limited  to 

•„  streams  having  slopes  within  that  range.     By 

-W-'  =  Z°-5  postulate, 


Then,  since  discharge  is  the  product  of  width, 
depth,  and  velocity, 


5_   T2.5  M141 


Q, 


F, 


Q,~w, 
whence 

and 


Substituting  this  value  of  the  fineness  factor  in 
(113),  and  reducing,  wo  have 


..  .(115) 


The  result  indicates  that 


whence 


- 

C,     Q, 


^ 
Q,,    Q, 


C 


or,  since  -Q=  U,  the  tractional  duty  of  water, 


That  is,  for  similar  streams  the  tractional  duty  of 
water  is  the  same. 

As  the  exponents  connecting  capacity  with 
discharge,  capacity  with  fineness,  and  mean 
velocity  with  hydrauh'c  radius  are  all  averages 
of  low  precision,  the  result  is  far  from  being  so 
secure  as  might  be  inferred  from  equation  (115). 
Its  best  support  is  really  found  in  the  plausi- 
bility of  its  conclusion.  Our  experience  with  a 
variety  of  physical  laws  makes  it  easy  for  us  to 
believe  that  with  suitable  parity  of  conditions 
a  unit  of  discharge  will  accomplish  the  same 
work  as  part  of  a  large  stream  that  it  will  ac- 
complish as  part  of  a  small  stream  ;  and  so  the 
conclusion  is  plausible.  The  fact  that  an  at- 
tempt to  test  the  hypothesis  of  similar  streams 
by  combining  it  with  experimental  data  has  led 
to  a  plausible  result  is  a  fact  favorable  to  the 
hypothesis. 

Let  us  now  assume  the  hypothetic  law  to  be 
a  real  law  and  draw  such  inferences  as  may  be 
warranted.  The  range  of  slopes  which  the  ex- 


A/"_ r ' ._  T 
D,  ~  V,,~ 

Substituting  in  equation  (114),  we  have 

ef~\E7. 

whence 


0.5 
1.0 
2.0 
3.0 


That  is  to  say,  for  similar  streams,  the  ratio  ^-5 

may  be  regarded  as  constant.  This  relation 
affords  a  criterion  for  the  discrimination  of 
those  natural  streams  which  are  similar  to  the 
laboratory  streams,  provided  they  are  also 
similar  in  slope  and  form  ratio.  The  following 

limiting  values  for  j^  for  different  slopes  are 

all  estimated  on  the  assumption  of  a  form  ratio 
of  0.05: 

Limiting  values  of     " 

3,000,000-40,000,000 
2,000,000-30,000,000 
1,500,000-10.000,000 
500,000-  4,000,000 

The  form  ratio  0.05  is  considerably  below  the 
average  of  the  ratios  developed  in  the  labora- 
tory, and  it  is  also  much  above  the  average 
for  alluvial  rivers  at  flood  stage.  Any  allow- 
ance which  might  be  made  for  this  discrepancy 
would  have  the  effect  of  increasing  the  estimate 
of  limiting  values  of  the  ratios  of  Q  to  .D2-5. 
Subject  to  this  qualification  the  ratios  indicate 
the  types  of  natural  streams  which  are  "similar" 
to  the  laboratory  streams  and  to  which  various 
laboratory  results  may  be  applied.  The  streams 
are  in  general  either  small  creeks  or  else  rivers 
transporting  very  coarse  debris.  As  the  slopes 
are  determined  by  flood  discharges,  such  dis- 
charges should  be  used  in  the  classification. 

For  the  streams  thus  classified  as  similar  to 
laboratory  streams  the  duty  of  water  is  of  the 
same  order  of  magnitude,  and  so  are  the  rates 
of  variation  of  duty  with  the  several  conditions 
of  slope,  discharge,  and  fineness.  The  rates  of 
variation  apply  especially  to  comparisons  of 
one  stream  with  another.  For  the  estimation 
of  variation  with  discharge  in  the  game  stream 
something  should  bo  added  to  the  laboratory 
rate  to  allow  for  the  varying  assistance  which 


240 


TRANSPORTATION    OF   DEEMS   BY   BUNKING   WAITER. 


suspension  gives  to  the  work  of  traction  (p.  234) . 
For  estimation  of  optimum  form  ratio  some- 
thing should  be  deducted  from  the  laboratory 
indication  to  allow  for  the  greater  resistance 
of  the  channel  walls. 

SUMMARY. 

Natural  streams  of  alluvial  type  differ  from 
the  streams  used  in  the  laboratory  in  ways  con- 
nected with  the  bondings  of  their  courses  and 
with  variations  of  discharge.  The  differences 
affect  forms  of  cross  section,  the  distribution  of 
velocities  within  the  section,  and  the  partition 
of  the  load  between  suspension  and  traction. 
The  two  portions  of  load  are  carried  at  the 
expense  of  the  stream's  energy,  each  reduces 
the  velocity,  and  the  reduction  of  velocity  de- 
termines the  limit  of  carrying  power.  The 
whole  burden  of  the  stream  includes  not  only 
two  divisions  distinguished  by  mode  of  trans- 
portation but  as  many  minor  divisions  as  there 
are  grades  of  debris,  and  the  load  carried  of 
each  grade  reduces  the  capacity  for  all  the 
grades. 

These  and  other  complexities  make  it  diffi- 
cult to  apply  the  laboratory  results  to  natural 
streams.  It  is  probable  that  the  forms  of  the 
laboratory  formulas  are  applicable,  with  limi- 
tations, to  the  comparison  of  one  stream  with 
another,  but  the  availability  of  the  exponents 
is  problematic.  There  are  special  difficulties  in 
attempting  to  use  the  formulas  for  the  compari- 
son of  capacities  of  the  same  stream  at  different 
stages,  and  in  such  comparisons  the  tractional 
load  can  not  be  considered  by  itself,  because 
much  material  "which  is  swept  along  the  bed 
at  lower  stages  is  lifted  by  flood  velocities  into 
the  body  of  the  current. 

It  is  thought  that  the  laboratory  formulas 
may  be  applied  to  natural  streams  which 
are  geometrically  similar  to  the  laboratory 
streams — -that  is,  to  streams  having  the  same 
slopes  and  form  ratios  and  carrying  d6bris  of 
proportionate  size.  The  class  of  streams  to 
which  the  formulas  apply  by  reason  of  simi- 
larity is  necessarily  restricted,  being  character- 
ized in  the  main  by  high  slopes  and  coarse 
debris.  It  can  include  few  large  streams. 

CONCLUSION. 

It  was  a  primary  purpose  of  the  Berkeley  in- 
vestigation to  determine  for  rivers  the  relation 


which  the  load  swept  along  the  bed  bears  to  the 
more  important  factors  of  control.  As  a  means 
to  that  end  it  was  proposed  to  study  the  mode  of 
propulsion  and  learn  empirically  the  laws  con- 
.nectirig  its  output  with  each  factor  of  control 
taken  separately.  The  review  of  results  in  the 
present  chapter  shows  that  the  primary  purpose 
was  not  accomplished.  In  the  direction  of  the 
secondary  purpose  much  more  was  achieved, 
and  a  body  of  definite  information  is  contrib- 
uted to  the  general  subject  of  stream  work.  A 
valuable  outcome  is  the  knowledge  that  the 
output  in  tractional  load  is  related  to  the  con- 
trolling conditions  in  a  highly  complex  manner, 
the  law  of  control  for  each  condition  being  qual- 
ified by  all  other  conditions. 

With  the  aid  of  the  Berkeley  experience  it 
would  be  possible  to  avoid  certain  errors  of 
method  and  arrange  experiments  which  should 
yield  more  accurate  measurements  of  the  same 
general  class — and  it  is  natural  that  the  experi- 
menter should  feel  the  desire  to  do  his  work  over 
in  a  better  way — but  I  am  by  no  means  sure  that 
adequate  advantage  would  reward  a  continu- 
ance of  work  on  the  same  or  closely  related  lines. 
The  complex  interactions  could  be  given  better 
numerical  definition,  but  it  may  well  be 
doubted  whether  their  empiric  definition  would 
lead  to  their  explanation.  It  is  possible  that 
the  chasm  between  the  laboratory  and  the  river 
may  be  bridged  only  by  an  adequate  theory, 
the  work  of  the  hydromechanist.  It  is  possible 
also  that  it  may  be  practically  bridged  by  ex- 
periments which  are  more  synthetic  than  ours, 
such  experiments  as  may  be  made  in  the  model 
rivers  of  certain  German  laboratories.  (See 
p.  16.) 

The  practical  applications  for  results  from 
experiments  in  stream  traction  belong  almost 
wholly  to  the  field  of  river  engineering.  For 
the  transportation  of  detritus  and  related  mate- 
rials by  artificial  currents,  stream  traction  will 
rarely  be  used,  because  flume  traction  is  more 
efficient.  Our  results  in  flume  traction  have 
therefore  an  immediate  practical  application, 
and  as  they  were  limited  in  range  the  advantage 
of  extending  them  can  hardly  be  questioned. 

The  report  on  the  Berkeley  investigation 
properly  closes  with  this  chapter,  but  there  are 
several  by-products  which  seem  worthy  of  rec- 
ord. Some  of  them  are  presented  in  the  fol- 
lowing chapter,  and  others  are  contained  in 
appendixes. 


CHAPTER  XIV.— PROBLEMS  ASSOCIATED  WITH  RHYTHM. 


RHYTHM  IN  STREAM  TRANSPORTATION. 

This  chapter  is  concerned  with  certain  prob- 
lems upon  which  the  Berkeley  investigation 
touched,  but  which  were  not  seriously  attacked. 

The  low  precision  of  the  observations  on 
stream  traction,  a  precision  characterized  by  an 
average  error  of  about  11  per  cent  and  an.  aver- 
age probable  error  after  adjustment  between  2 
and  3  per  cent,  had  for  its  chief  cause  the  failure 
to  eliminate  from  the  experiments  the  influ- 
ence of  rhythm.  The  slope  of  the  water  sur- 
face, the  slope  of  the  channel  bed,  and  the  load 
of  debris  transported  were  all  subject  without 
intermission  to  rhythmic  fluctuations.  If  ma- 
terially better  observations  of  the  same  sort  are 
to  be  made,  this  difficulty  must  be  successfully 
dealt  with,  and  the  first  step  toward  mastering 
it  is  to  understand  it.  The  removal  of  a  diffi- 
culty, however,  is  neither  the  sole  nor  the  most 
important  result  to  be  expected  from  the  study 
of  fractional  rhythms.  Underlying  them  are 
physical  principles  which  are  of  importance  in 
the  dynamics  of  rivers,  and  their  study  con- 
stitutes one  of  the  available  lines  of  approach 
to  the  broader  subject. 

For  their  empiric  study  the  general  plan  of 
the  Berkeley  apparatus  is  well  adapted,  but  our 
experience  indicates  that  certain  details  should 
be  modified.  The  use  of  a  long  trough  is  ad- 
visable, with  contraction  at  the  outfall,  and 
with  delivery  of  the  load  to  a  settling  tank  be- 
yond the  outfall.  The  appliances  and  methods 
should  be  such  as  to  secure  uniformity  in  dis- 
charge, in  character  of  d6bris,  and  in  rate  of 
feed. 

The  apparatus  for  regulating  discharge,  de- 
scribed on  pages  20  and  257,  was  one  of  the  most 
satisfactory  parts  of  the  Berkeley  equipment. 
Its  most  important  feature,  as  affecting  pre- 
cision, was  the  delivery  of  the  water  through  an 
aperture  under  a  considerable  head. 

Uniformity  of  debris  can  hardly  be  secured 
without  the  employment  of  an  artificial,  nar- 
rowly limited  grade,  and  the  available  means 
of  sorting  is  the  sieve.  It  is  to  be  observed, 
however,  that  after  a  grade  has  been  separated 


by  sieves  it  is  still  subject  to  sorting  by  current, 
the  current  recognizing  differences  of  form  and 
density  which  the  sieves  ignore.  When  d6bris 
that  has  once  been  handled  by  the  stream  is  to 
be  used  a  second  time,  remixing  may  be  ad- 
visable. 

None  of  the  devices  we  employed  to  feed 
debris  to  the  current  achieved  uniformity. 
Those  which  depended  on  the  flow  of  wet  de- 
bris through  an  aperture  failed  because  the 
proportion  of  water  could  not  be  kept  constant. 
The  others  depended  on  handwork  and  ex- 
perienced the  irregularity  usual  to  handwork. 
An  apparatus  planned  near  the  end  of  the  ex- 
perimental work,  but  never  tried,  is  of  such 
promise  that  its  essential  features  are  here 
described. 


20921°— No.  86—14- 


-16 


FIGPBE  72.— Suggested  apparatus  for  automatic  feed  of  debris. 

A  drum,  A  in  figure  72,  is  turned  slowly  by 
power,  its  rate  being  regulated  by  clockwork. 
Its  position  is  above  the  trough  containing  the 
experiment  stream,  B.  The  surface  of  the 
drum  is  uniformly  roughened.  Above  it  is  a 
vertical  rectangular  shaft,  filled  with  moist 
debris.  The  shaft  does  not  touch  the  drum. 
The  width  of  the  separating  space  at  D  is  con- 
trolled by  some  suitable  device.  As  the  drum 
turns,  a  debris  layer  of  uniform  thickness  is 
carried  with  it,  and  this  falls  into  the  stream. 
To  prevent  irregularities  due  to  adhesion,  a 

241 


242 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


detaching  device  is  placed  at  some  point  E, 
the  device  possibly  consisting  of  an  open-rank 
comb  of  elastic  wire.  The  shaft  should  be 
smooth  and  of  uniform  section,  so  that  the 
particles  of  de'bris  near  the  drum  may  be 
brought  into  actual  contact  with  one  another 
by  pressure  of  the  debris  above.  That  the 
debris  may  not  be  caused  to  flow  by  excess  of 
moisture,  it  should  be  thoroughly  drained  be- 
fore use.  In  a  rough  construction  designed  to 
test  the  practicability  of  the  apparatus  the 
drum  surface  was  roughened  by  covering  with 
a  wire  screen,  and  this  was  found  to  secure  the 
delivery  of  the  de'bris. 

Uniformity  of  feed  having  been  provided,  the 
rhythms  of  transportation  may  be  observed  as 
oscillations  in  the  de'bris  delivered  at  the  out- 
fall. Rhymths  of  slope  may  be  studied,  at 
least  initially,  by  observing  changes  in  the 
profile  of  the  water  surface. 

In  the  record  of  the  Berkeley  observations  it 
is  not  practicable  fully  to  discriminate  rhyth- 
mic inequalities  from  those  occasioned  by 
irregularities  of  de'bris  feeding,  but  there  is 
reason  to  believe  that  several  rhythms  of  differ- 
ent period  coexist.  The  shortest  rhythms  are 
those  connected  with  the  dunes  and  antidunes, 
and  these  are  evidently  associated  with  rhythms 
of  the  flow  of  water. 

RHYTHM  IN  THE  FLOW  OF  WATER. 

Reynolds,1  treating  of  the  flow  of  water 
through  tubes,  distinguishes  two  modes  of  flow 
as  direct  and  sinuous.  They  are  otherwise 
called  steady  and  turbulent.  In  direct  flow  the 
filaments  of  current  have  simple  lines,  which 
are  straight  and  parallel  if  the  walls  of  the  con- 
duit are  straight  and  parallel.  In  sinuous  flow 
the  filaments  of  current  are  neither  simple  nor 
parallel  and  may  be  intricately  convoluted. 
The  flow  in  tubes  is  direct  for  low  velocities 
and  sinuous  for  high,  the  critical  velocity  vary- 
ing inversely  with  the  diameter  and  the  rough- 
ness of  the  tube  and  directly  with  the  viscosity 
of  the  water.  It  would  follow  from  his  gener- 
alizations that  the  flow  of  such  streams  as  were 
used  in  our  experiments  would  be  sinuous,  and 

i  Reynolds,  Osborne,  An  experimental  investigation  of  the  circum- 
stances which  determine  whether  the  motion  of  water  shall  be  direct  or 
sinuous  and  of  the  law  of  resistance  in  parallel  channels:  Eoy.  Soc. 
I'hilos.  Trans.,  London,  vol.  174,  pp.  935-982,  1883.  Also,  The  two 
manners  of  motion  of  water:  Roy.  Inst.  Great  Britain  Proc.,  1884. 


its  actual  sinuosity  was  a  matter  of  observa- 
tion. To  a  large  extent  the  curvature  of  the 
flow  lines  was  shown  by  the  motions  of  minute 
suspended  particles,  and  when  this  evidence 
was  lacking  it  was  still  possible  to  infer  diver- 
sity of  current  from  continual  changes  in  the 
configuration  of  the  water  surface. 

It  is  probable  that  all  the  diversities  of  flow 
were  rhythmic,  for  that  is  the  nature  of  in- 
equalities of  motion  developed  by  the  inter- 
action of  constant  forces;  but  in  many  cases 
the  rhythms  were  so  numerous  and  so  related 
in  period  and  other  characters  that  their  com- 
bination gave  the  impression  of  irregularity. 
In  many  other  cases,  however,  some  one 
rhythm  was  dominant,  and  these  cases  appear 
especially  worthy  of  study. 

The  cycle  of  movements  constituting  a 
rhythm  unit  may  be  definitely  related  to  space, 
or  to  time,  or  to  both  space  and  time.  The 
pulsations  of  the  water  surface  which  ob- 
structed our  observations  of  water  profile  were 
manifestations  of  time  factors.  The  regular 
sequence  of  dune  crests  was  a  manifestation 
of  a  space  factor.  The  two  phenomena  co- 
existed. In  the  space  interval  from  dune  crest 
to  dune  crest  certain  elements  of  motion  were 
constant.  There  was  a  large  stationary  vortex 
in  the  hollow  between  the  crests  (fig.  10,  p.  31), 
and  there  might  be  stationary  elements  in  the 
configuration  of  the  water  surface.  At  the 
same  time  the  flow  lines  of  the  water  at  all 
points  swayed  in  direction  and  were  affected 
by  variations  of  velocity,  and  these  changes 
were  rhythmic  with  respect  to  time.  Doubt- 
less many  of  the  fluctuations  belonged  to  or 
were  associated  with  vortices  which  traveled 
with  the  general  current.  It  is  conceivable, 
or  even  probable,  that  the  stationary  features 
and  the  traveling  features  were  coordinated,  so 
that  within  the  rhythmic  space  unit  corres- 
ponding to  the  dune  interval  there  was  a 
cycle  of  variations  of  motion  which  was 
rhythmic  with  respect  to  time. 

The  rhythmic  dunes  marched  slowly  down  the 
trough  and  with  them  marched  the  intercrest 
vortices  and  associated  motions.  To  that  ex- 
tent the  features  of  the  bed  controlled  the  fea- 
tures of  the  current.  Nevertheless  the  dunes 
were  not  essential  to  the  existence  of  a  water 
rhythm  characterized  by  a  definite  space  inter- 
val. There  was  positive  evidence  that  the  dune 


PEOBLEMS   ASSOCIATED   WITH    RHYTHM. 


243 


interval  was  determined  by  a  preexistent  water 
rhythm.  Xot  only  was  the  dune  interval  a 
function  of  depth  and  velocity  of  current,1  but 
the  creation  of  dunes  by  water  rhythm  was 
repeatedly  observed.  In  certain  experiments  a 
slow  current,  moving  over  a  bed  of  debris  arti- 
ficially smoothed  and  leveled,  was  gradually 
quickened  until  transportation  began.  The 
movement  of  particles  did  not  begin  at  the  same 
time  all  along  the  bed  but  was  initiated  in  a 
series  of  spots  separated  by  uniform  intervals, 
and  the  first  result  of  the  transportation  was  the 
creation  of  a  system  of  dunes. 

In  certain  experiments  on  flume  traction  a 
slow  current,  moving  over  a  smooth  channel 
bed  of  wood,  swept  along  a  small  quantity  of 
sand.  With  increase  of  the  load  of  sand  local 
deposits  were  induced,  which  took  the  form  of 
thin  straggling  patches,  similar  to  one  another 
in  outline  and  separated  by  approximately 
equal  bare  spaces.  These  moved  slowly  down- 
stream, the  mode  of  progress  being  similar  to 
that  of  dunes,  and  with  further  increase  of  load 
they  acquired  the  typical  profile  of  dunes. 

In  both  groups  of  experiments  it  was  evident 
that  the  primary  rhythm  pertained  to  the  water 
and  was  independent  of  the  work  of  transpor- 
tation. In  the  second  group,  where  sand  swept 
steadily  forward  over  the  whole  area  of  the  bed, 
it  was  evident  that  the  water  rhythm  did  not 
involve  reversed  currents  along  the  bed  and 
therefore  did  not  include  such  stationary  vor- 
tices as  accompany  dunes. 

In  the  experiments  on  stream  traction  the 
development  of  dunes  was  conditioned  by  the 
three  factors  of  velocity,  depth,  and  load,  be- 
sides an  undetermined  influence  from  fineness 

i  Measurements  of  dune  interval  in  the  laboratory  were  too  few  to  dem- 
onstrate the  factors  of  control,  but  comparison  of  laboratory  data  with 
data  from  other  sources  leaves  little  question  that  control  is  exercised  by 
depth,  velocity,  and  fineness.  In  the  laboratory,  where  depths  were  a 
matter  of  inches,  the  dune  interval  rarely  exceeded  2  feet.  In  Mississippi 
River  depths  measured  in  scores  of  feet  are  associated  with  intervals 
measured  in  hundreds  of  feet.  In  tidal  estuaries,  where  dunes  of  a 
special  type  are  exposed  at  low  tide,  the  depths  of  the  formative  cur- 
rents are  intermediate  between  those  of  the  laboratory  and  those  of  the 
Mississippi,  while  the  intervals  are  measured  in  feet  or  tens  of  feet.  For 
data  of  the  Mississippi  see  Johnson,  J.  B.,  Kept.  Chief  Eng.  U.  8.  A., 
1879,  pp.  1963-1967,  and  Eng.  News,  1885,  pp.  68-71;  Powless,  W.  H., 
Mississippi  River  Comm.  Kept,  for  1881,  pp.  66-120;  and  especially 
Hider,  Arthur,  Mississippi  River  Comm.  Rept.  for  1882,  pp.  83-88.  For 
data  on  the  dunes  of  tidal  estuaries  see  Cornish,  Vaughan,  Geog.  Jour., 
vol.  18,  pp.  170-202,  1901.  Hider  finds  that  the  dune  interval  is  greater 
at  high  stages  of  the  river  than  at  low,  the  depth  and  velocity  both 
decreasing  with  the  change  from  high  to  low.  Cornish's  observations 
show  that  the  interval  varies  directly  with  depth  of  current  under  condi- 
tions which  make  it  probable  that  the  velocity  varies  inversely  with 
depth. 


of  debris.2  It  is  probable  that  load  and  fine- 
ness enter  only  as  factors  of  resistance,  so  that 
the  essential  conditions  are  velocity,  depth,  and 
bed  resistance.  Within  certain  rather  wide 
ranges  of  value  for  these  controlling  factors  the 
bed  is  molded  into  dunes.  When  the  limit  is 
exceeded  by  increase  of  velocity  or  resistance, 
or  by  decrease  of  depth,  the  dunes  disappear 
and  the  bed  becomes  smooth  and  plane.  At 
the  same  time  the  oscillations  and  other  dis- 
turbances of  the  water  surface  are  reduced;  but 
as  they  do  not  altogether  disappear  it  is  to  be 
inferred  that  the  flow  is  still  characterized  by 
internal  diversity. 

With  still  further  increase  of  velocity  or 
resistance,  or  with  further  reduction  of  depth, 
another  critical  point  is  passed,  and  the  process 
of  traction  becomes  again  rhythmic,  but  in  an 
antithetic  way.  The  bed  is  molded  into  anti- 
dunes,  which  travel  upstream  (pp.  31-34), 
and  the  water  surface  also  is  molded  into 
waves,  which  copy  the  forms  of  the  antidunes 
and  move  with  them.  The  internal  move- 
ment of  the  water  is  again  characterized  by  a 
dominant  rhythm,  but  the  type  of  rhythm  is 
different  from  that  associated  with  dunes. 
The  rhythm  is  also  less  stable,  and  its  intensity 
exhibits  a  cycle  of  change.  With  low  intensity 
the  waves  are  nearly  equal  in  height  and 
length,  but  sooner  or  later  inequalities  develop 
and  the  higher  waves  overtake  the  lower  and 
absorb  them.  This  process  increases  the  wave 
length  in  the  upstream  part  of  the  trough,  and 
the  influence  of  the  change  is  hi  some  way 
communicated  to  the  downstream  end,  where 
the  waves  are  first  formed,  with  the  result 
that  larger  and  larger  waves  develop.  Finally 
a  master  wave,  with  curling  crest,  rushes 
through  the  trough  from  end  to  end,  and  this 
has  the  effect  of  wiping  out  the  irregularities 
and  restoring  the  status  of  low  intensity. 
Various  phases  of  the  cycle  are  illustrated  by 
the  profiles  in  figure  12  (p.  33). 

When  combinations  of  velocity,  resistance, 
and  depth  similar  to  those  causing  antidunes 
are  made  for  a  stream  flowing  through  a  rigid 
straight  channel,  without  movable  debris,  the 
water  develops  surface  waves,  and  these  travel 

2  Eger,  Dix,  and  Seifert  (Zeitschr.  Bauwesen,  vol.  56,  pp.  325-328) 
found  that  under  certain  conditions  dunes  were  developed  in  a  sand  of 
uniform  grade  but  not  in  a  finer  sand  composed  of  several  grades.  In 
our  experiments  the  smooth  phase  of  transportation  had  greater  range 
with  mixed  grades  than  with  single  grades. 


244 


TRANSPOKTATION   OF   DEBKIS  BY   SUNNING   WATEE. 


downstream  faster  than  the  current.  They 
are  initially  rhythmic,  but  their  period  is 
unstable  because  their  velocity  of  propagation 
varies  with  their  size.  A  wave  with  slight 
advantage  in  size  will  overtake  the  one  in 
front  of  it,  and  then  the  two  will  unite,  making 
a  wave  with  still  higher  velocity.  Thus  the 
system  tends,  for  a  time  at  least,  toward 
reduction  of  the  number  of  waves  and  increase 
of  the  wave  interval.  As  the  waves  grow  by 
composition,  their  fronts  steepen  and  the 
culminating  phase  is  that  of  a  "roll  wave,"  or 
bore.1  It  may  be  noted,  in  passing,  that  the 
pulsations  frequently  observed  in  the  overfall 
of  a  dam  are  probably  rhythms  of  this  type. 

It  may  be  assumed,  at  least  tentatively, 
that  all  the  dominant  rhythms  observed  in 
straight  conduits  are  initiated  at  the  intake. 
Those  associated  with  low  velocity  and  an 
immobile  bed  appear  to  be  stationary,  but 
with  a  mobile  bed  they  develop  dunes,  and 
they  then  travel  downstream  with  the  dunes. 
Those  associated  with  high  velocity  and  an 
immobile  bed  develop  waves  of  translation 
which  travel  downstream.  With  a  mobile  bed 
antidunes  are  developed,  and  surface  waves 
travel  with  them  upstream.  It  is  possible 
that  the  antidune  waves  coexist  with  the 
waves  of  translation,  and  that  the  cycle  of 
intensity  in  the  antidune  phenomena  results 
from  the  interactions  of  the  two  systems. 

If  there  is  warrant  for  the  various  correlations 
above  indicated,  the  phenomena  developed  by 
our  experiments  in  connection  with  modes  of 
debris  transportation  afford  a  basis  of  classifi- 
cation for  a  considerable  body  of  water 
rhythms.  What  we  have  called  the  smooth 
phase  of  traction  marks  a  critical  phase  of 
water  flow  separating  two  types  which  are 
characterized  by  dominant  rhythms,  and  the 
two  dominant  rhythms  are  in  some  way  anti- 
thetic. The  nature  of  their  antithesis  is  not 
known  to  me,  nor  is  the  character  of  either 
rhythm;  but  it  appears  that  the  experimental 
and  analytic  study  of  the  rhythms  constitutes 
a  field  of  research  which  is  at  the  same  time 
promising  and  important.  I  assume  that  the 
definite  rhythms  of  water  by  which,  when 
debris  is  present,  the  dunes  and  antidunes  are 
caused  are  susceptible  of  analytic  treatment, 

1  See  Cornish,  Vaughan,  Progressive  wfcves  in  rivers:  Geog.  Jour. 
(London),  vol.  29,  pp.  23-31, 1907. 


and  I  believe  that  the  experiment  trough 
affords  the  means  of  preliminary  delineation 
and  ultimate  verification. 

THE  VERTICAL  VELOCITY  CURVE. 

No  systematic  study  was  made  of  the  distri- 
bution of  velocities  within  the  streams  of  the 
laboratory,  but  incidentally,  in  connection  with 
several  different  minor  inquiries,  the  vertical 
distribution  of  velocities  in  the  medial  plane 
was  recorded;  and  it  happens  that  some  of 
these  records  serve  to  illustrate  the  depend- 
ence of  that  distribution — the  vertical  velocity 
curve— on'  certain  conditions.  The  observa- 
tions were  made  with  the  Pitot-Darcy  gage. 

The  first  to  be  reported  were  made  ia  con- 
nection with  a  comparison  of  the  free  outfall 
(fig.  1,  p.  19)  and  the  contracted  outfall  (fig.  3, 


FIGURE  73. — Modification  of  vertical  velocity  curve  by  approach  to 
outfall. 

p.  25).  The  trough  used  with  free  outfall  had 
smooth  sides  and  bed,  was  1  foot  wide,  and 
was  set  level.  The  stream  of  water,  0.734 
ft.3/sec.,  carried  no  debris.  Its  surface  slope 
gradually  increased  toward  the  outfall,  and  the 
mean  velocity  increased  with  the  fall  of  the 
surface.  Velocities  were  measured  at  three 
points,  respectively,  5.5  feet,  2.3  feet,  and  0.8 
foot  from  the  outfall.  Their  curves,  given  in 
figure  73,  show  that  the  acceleration  as  well  as 
the  velocity  increased  as  the  outfall  was  ap- 
proached, and  that  there  was  a  coordinate 
modification  of  the  shape  of  the  curve.  In  the 
first  curve  it  is  evident,  despite  a  discordance 
among  the  determined  points,  that  the  level  of 
maximum  velocity  is  somewhat  above  mid- 
depth;  in  the  second  the  maximum  is  near  mid- 
depth;  and  in  the  third  it  is  below.  The  accel- 
eration, for  Avhich  an  integrated  expression 
appears  in  the  horizontal  space  between  curves, 


PROBLEMS   ASSOCIATED   WITH   BHYTHM. 


245 


seems  to  have  affected  the  lower  part  of  the 
current  more  than  the  upper. 

The  trough  used  with  contracted  outfall  was 
0.92  foot  wide  and  was  contracted,  in  the 
manner  shown  in  figure  3  (p.  25)  to  0.30  foot. 
The  oblique  walls  producing  the  contraction 


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Velocity 

FIGURE  74. — Modification  of  vertical  velocity  curve  by  approach  to 
contracted  outfall. 

were  3  feet  long.  The  first  velocity  station 
was  1.5  feet  from  the  outfall,  at  a  point  where 
the  width  was  0.64  foot;  the  second  and  third 
were  at  0.75  and  0.25  foot  from  the  outfall, 
corresponding  to  widths  of  0.47  and  0.35  foot. 
The  Velocity  curves,  given  in  figure  74,  show 
the  descent  of  the  plane  of  maximum  velocity 
from  a  position  slightly  above  middepth  to 


FIGURE  75. — 1'lan  of  experiment  trough  with  local  contraction.  The 
letters  show  stations  at  which  velocities  were  observed,  and  are  re- 
peated in  figures  7G  and  77.  Scale,  1  inch«=2  feet. 

near  the  bottom.  The  acceleration,  in  this 
case  connected  with  contraction  of  channel, 
as  well  as  with  the  release  at  outfall  from  the 
channel  resistance,  is  a  function  of  depth. 

In  the  third  arrangement  a  trough  0.92  foot 
wide,  with  a  slope  of  0.58  per  cent,  was  con- 
tracted at  one  point  in  the  manner  indicated 
in  figure  75.  This  gave  to  the  stream  such  a 


FIGURE  76. — Profile  of  waU-r  surface  in  trough  shown  in  figure  75.    Scale 
1  inch=2feet. 

profile  as  is  sketched  in  figure  76.  Velocities 
were  measured  at  four  points,  the  distances 
from  the  point  of  extreme  contraction  being 
2.0  feet,  1.0  foot,  0.5  foot,  and  0,  and  the  corre- 
sponding widths  of  current  0.92,  0.55,  0.37, 
and  0.19  foot.  The  velocity  curves  are  plotted 


in  figure  77.  The  level  of  maximum  velocity 
is  at  or  near  the  surface  at  the  point  of  initial 
contraction,  A,  and  then  drops  quickly,  being 
below  0.9  depth  in  the  narrowest  strait,  D. 
In  this  case  also  the  acceleration  appears  to 
increase  in  regular  manner  with  distance  from 
the  surface. 

To  avoid  the  peculiarities  observed  near  the 
outfall,  as  well  as  those  appropriate  to  intake 


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Velocity 

4 

FIGTOE  77. — Modification  of  vertical  velocity  curve  by  local  contrac- 
tion of  channel.  The  letters  indicate  positions  of  velocity  stations  in 
trough.  (See  fig.  75.) 

conditions,  all  other  determinations  of  the 
curve  were  made  near  midlength  of  the  trough, 
the  ordinary  distance  from  the  outfall  being 
17  feet  and  the  least  distance  13  feet. 

In  figure  78  are  three  curves  illustrating 
the  influence  of  slope  of  channel.  The  trough 
was  1.96  feet  wide  and  its  sides  and  bed 
were  smooth;  the  discharge  was  1.119ft.3/sec. 
Curve  A  shows  velocities  17  feet  from  the  out- 


FiGURE  78. — Modification  of  vertical  velocity  curve  when  mean  velocity 
is  increased  by  change  of  slope. 

fall,  with  the  trough  level;  B  corresponds  to  a 
slope  of  0.26  per  cent  and  C  to  0.56  per  cent. 
With  the  trough  level,  maximum  velocity 
occurs  at  about  0.75  depth,  and  with  both 
inclinations  of  trough  its  position  is  indicated 
at  from  0.2  depth  to  0.3  depth.  As  curve  0 
differs  in  type  from  all  others  obtained,  and  as 
it  may  be  made  harmonious  by  rejecting  a 
single  observation,  it  is  thought  probable  that 
the  dotted  line  better  expresses  the  fact.  Its 


246 


TRANSPOKTATION   OF   DEBEIS   BY   KUNNING   WATEE. 


substitution  throws  the  maximum  to  the  sur- 
face of  the  water  and  makes  a  consistent  series 
of  the  three  curves. 

In  another  series  of  observations  the  trough 
remained  horizontal  while  the  discharge  was 


.923 


Velocity 

FIGURE  79. — Modification  of  vertical  velocity  curve  when  mean  velocity 
is  increased  by  change  of  discharge.  The  corresponding  discharges 
are  indicated  in  ft.'/sec. 

varied.  The  width  was  0.66  foot,  and  the  dis- 
charges 0.182,  0.363,  0.545,  0.734,  and  0.923 
ft.3/sec.  The  resulting  curves  (fig.  79),  with 
exception  of  that  for  the  smallest  discharge, 
form  a  consistent  series,  the  level  of  maximum 
velocity  rising  slowly  with  increase  of  discharge. 
In  this  case  also  the  discordant  curve  may  be 
brought  into  harmony  by  the  rejection  of  a 


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Velocity 

1'iGUKE  80. — Modification  of  vertical  velocity  curve  by  changes  in  the 
roughness  of  the  channel  bed.  Curve  1,  paraffin;  2,  roughness  of  debris 
grade  (A);  3,  roughness  of  grade  (D);  4,  roughness  of  grade  (F). 

single  observation,  and  a  substitute  curve  is 
suggested  in  the  diagram  by  a  broken  line.1 

In  another  series  the  texture  of  the  channel 
bed  was  made  to  vary.  Constant  features  were 
the  discharge,  0.734  ft.3/sec.;  the  width,  0.92 
foot;  the  slope,  0.58  per  cent;  and  the  texture 
of  the  channel  sides,  which  were  planed  and 
unpainted.  The  varieties  of  bed  texture  were 
(1)  paraffin,  coating  a  smooth  board;  (2)  a 

'  In  this  instance,  in  the  one  before  mentioned,  and  also  in  the  case  of 
three  curves  in  figure  80  the  aberrant  point  records  a  measurement  which 
was  made  very  near  the  surface  of  the  water.  In  such  positions  the 
constant  of  the  Pitot-Darcy  gage  has  a  special  value,  and  it  is  on  the 
whole  probable  that  the  apparent  errors  of  observation  are  occasioned 
by  an  error  of  the  rating  formula.  The  matter  is  discussed  in 
Appendix  A. 


pavement  of  debris  of  grade  (A);  and  (3  and  4) 
similar  pavements  with  grades  (D)  and  (F). 
The  observed  curves,  given  in  figure  80,  indi- 
cate that  the  resistance  occasioned  by  a  rough 
bed  retards  the  whole  current  but  retards  the 
lower  parts  in  greater  degree  than  the  upper, 
so  that  the  level  of  maximum  velocity  is  raised. 
The  amount  of  retardation  is  greater  as  the 
texture  of  the  bed  is  coarser. 

In  figure  81  are  two  curves  illustrating  the 
influence  of  load  on  the  vertical  distribution  of 
velocity.  One  curve  was  observed  in  a  stream 
without  load,  flowing  over  a  smooth  and  hori- 
zontal bed,  the  other  in  the  same  stream  when 
carrying  a  load  of  17  gm./sec.  on  a  self-adjusted 


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Velocity 

FIGURE  81. — Modification  of  vertical  velocity  curve  by  addition  of  load 
to  stream,  with  corresponding  increase  of  slope.  A,  without  load;  B 
with  load. 


slope  of  0.64  per  cent.  To  judge  from  the  data 
shown  in  figure  78  (p.  245),  the  effect  of  slope 
alone  would  be  to  double  or  nearly  double  the 
mean  velocity,  but  the  actual  increase  was  only 
13  per  cent.  The  acceleration  due  to  slope  was 
almost  wholly  neutralized  by  the  retardation 
due  to  roughness  of  bed  and  to  the  work  of 
traction.  The  retardation  had  also  the  effect 
of  raising  the  level  of  maximum  velocity. 


FIGUBE  82. — Modification  of  vertical  velocity  curve  by  addition  and 
progressive  increase  of  load. 

Similar  contrasts  are  shown  in  figure  82, 
where  the  curve  for  an  unloaded  stream  is  com- 
pared with  curves  for  the  same  stream  when 
transporting  three  different  loads.  The  con- 
stant factors  in  this  case  are:  Width  of  channel, 


PROBLEMS   ASSOCIATED   WITH   BHYTHM. 


247 


0.66  foot;  discharge,  0.545  ft.3/sec.;  grade  of 
debris,  (C).  The  variables  are  as  follows: 

Load gm./sec          0  38  53  194 

Slope  of  bed... per  cent..       0         0.38       0.5G         1.14 

Depth  at  17  feet  from  out- 
fall  feet..  0.353  0.313  0.303  0.226 

Mean  velocity...  ft. /sec..    2.57       2.64       2.73        3.  6G 

Level  of  maximum  veloc- 
ity; measured  from  the 
surface  as  a  fraction  of 
depth 0.7  0.1  0  0 

Letter  indicating  curve  in 
figure82 IB  C  D 

With  the  slope  of  0.38  per  cent  the  sand  moved 
in  dunes;  with  0.56  per  cent  the  phase  of  trac- 
tion was  transitional  between  the  dune  and  the 
smooth;  with  1.14  it  was  transitional  between 
the  smooth  and  the  antidune.  The  conspicu- 
ous change  associated  with  the  addition  of  load 
is  the  raising  of  the  level  of  maximum  velocity, 
and  this  is  correlated  also  with  increase  of  slope, 
increase  of  mean  velocity,  decrease  of  depth, 
and  modification  of  the  mode  of  traction. 

The  plotted  points  for  velocities  near  the 
channel  bed  are  irregular  when  the  observations 
were  made  above  a  bed  of  loose  debris,  and 
little  use  has  been  made  of  them  in  drawing  the 
curves.  As  previously  mentioned,  the  presence 
of  the  gage  caused  a  deflection  of  the  lines  of 
flow  and  the  formation  of  a  hollow  in  the  bed. 
Not  only  was  it  impossible  to  observe  with 
accuracy  the  relation  of  the  instrument  to  the 
normal  position  of  the  bed,  but  the  velocity 
observed  was  higher  than  that  normally  asso- 
ciated with  the  depth  at  which  the  instrument 
was  placed.  (See  Appendix  A.) 

In  most  of  the  groups  of  curves  the  variations 
of  form  are  associated  with  simultaneous  varia- 
tions of  so  many  conditions  that  the  nature  of 
the  control  is  not  evident.  For  satisfactory 
interpretation  a  fuller  series  of  observations 
seems  to  be  required,  but  certain  inferences  may 
be  drawn  from  those  before  us. 

Many  of  the  peculiarities  of  form  are  con- 
nected with  the  position  of  the  level  of  maximum 
velocity.  The  movements  of  the  maximum  in 
relation  to  depth  of  current  are  of  two  kinds.  It 
rises  with  increase  of  depth  when  that  increase 
is  caused  by  increase  of  discharge  (fig.  79).  It 
falls  with  increase  of  depth  when  that  increase  is 
independent  of  discharge  (figs.  78,  81,  and  82). 
Apparently  depth,  considered  by  itself,  is  not 
a  factor  of  control.  The  maximum  rises  with 
increase  of  mean  velocity  when  that  increase  is 


due  to  increase  of  discharge  (fig.  79),  or  of 
slope  (fig.  78),  but  falls  with  increase  of  mean 
velocity  when  the  increase  is  due  to  lessened 
resistance  of  the  channel  bed  (figs.  73,  80,  81, 
and  82).  Apparently  mean  velocity,  consid- 
ered by  itself,  is  not  a  factor  of  control.  If  we 
give  attention  to  the  three  factors  on  which 
depth  and  mean  velocity  chiefly  depend — 
namely,  discharge,  slope,  and  bed  resistance — 
a  more  consistent  relation  is  found.  Variations 
of  discharge  affect  only  the  group  of  curves  in 
figure  79,  and  there  the  maximum  rises  with 
increase  of  discharge.  Slope  affects  the  groups 
in  figures  78,  81,  and  82,  and  in  each  case  the 
maximum  rises  with  increase  of  slope.  Bed 
resistance  affects  the  groups  in  figures  80,  81, 
and  82,  and  in  each  case  the  maximum  rises 
with  increase  of  resistance. 

The  lowering  of  the  level  of  maximum  veloc- 
ity as  the  point  of  outfall  is  approached  (fig.  73) 
is  a  harmonious  feature,  but  in  that  case  there 
is  substituted  for  progressive  reduction  of  bed 
resistance  an  abrupt  cessation  of  all  channel 
resistance.  The  resulting  acceleration  is  propa- 
gated upstream,  and  its  amount  has  a  vertical 
distribution  connected  with  pressure.  In  the 
case  of  contracted  outfall  (fig.  74),  there  is 
added  an  effect  of  convergence,  which  still  fur^ 
ther  illustrates  the  graduation  of  acceleration 
in  relation  to  pressure.  The  influence  of  con- 
traction is  important  in  other  connections,  but 
need  not  be  further  discussed  in  this  place. 

The  influence  of  outfall  may  extend  to  a 
considerable  number  of  the  curves  here  figured. 
In  most  of  the  experiments  made  without 
contraction  at  outfall  there  was  progressive 
decrease  of  depth  and  increase  of  mean 
velocity,  from  some  point  near  the  head  of  the 
trough  to  its  end.  This  was  most  marked 
when  the  bed  of  the  trough  was  horizontal 
(figs.  73,  78-4,  79,  81-4,  and  82-4).  Reasoning 
from  the  observed  fact  that  acceleration  in- 
creases with  depth,  I  think  it  probable  that 
under  such  conditions  the  level  of  maximum 
velocity  lies  lower  than  it  would  with  a  uniform 
mean  velocity. 

Returning  to  the  consideration  of  discharge, 
slope,  and  resistance,  we  may  note  that  the 
variable  resistance  with  which  variations  of 
the  curve  have  been  definitely  correlated  is  bed 
resistance.  In  all  the  experiments  the  channel 
sides  had  the  same  texture,  so  that  the  side 
resistance  was  approximately  proportional  to 


248 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


the  depth.  The  level  of  maximum  velocity 
thus  has  the  same  relation  to  side  resistance  as 
to  depth;  it  sometimes  rises  and  sometimes 
falls  when  side  resistance  is  increased.  While 
a  probability  exists  that  side  resistance  influ- 
ences the  position  of  the  maximum,  the  nature 
of  its  influence  is  not  shown  by  the  observations 
under  consideration. 

Discharge  and  slope,  or  the  energy  factors 
which  they  help  to  measure,  urge  the  water 
forward,  and  their  influence  is  applied  to  the 
whole  stream.  Bed  resistance  holds  the  stream 
back  but  is  applied  to  its  base  only.  The 
obvious  tendency  of  these  forces  is  to  make  the 
upper  part  of  the  stream  move  faster  than  the 
lower  and  produce  a  velocity  curve  with  maxi- 
mum at  the  water  surface.  This  tendency  is 
opposed  by  some  other  factor,  unknown,  which 
tends  to  depress  the  level  of  the  maximum. 
Whatever  that  other  factor  may  be,  it  loses  in 
relative  importance  when  discharge,  slope,  or 
bed  resistance  is  increased. 

A  noteworthy  feature  of  the  curves  is  a 
tendency  to  change  in  character  near  the  bed. 
The  observations  are  not  so  precise  nor  so  full 
as  to  afford  a  distinct  characterization  of  the 
change,  but  there  can  be  little  question  of  its 
existence.  It  would  appear  that  the  peculiar 
conditions  near  the  bed  give  great  local  impor- 
tance to  some  factor  of  velocity  control  which 
is  elsewhere  of  minor  importance.  Attention 
may  also  be  directed  to  the  fact  that  none  of 
the  curves  resembles  an  ordinary  parabola 
with  horizontal  axis.  Had  these  been  the 
vertical  velocity  curves  to  which  mathematical 
formulas  were  first  fitted,  the  equation  of  the 
parabola  would  not  have  been  used. 

The  statement,  above,  that  the  factor 
tending  to  depress  the  level  of  maximum 
velocity  below  the  water  surface  is  unknown 
is  perhaps  rash,  for  several  theories  as  to  its 
nature  have  been  advanced  with  confidence. 
To  put  the  matter  more  cautiously — it  has 
seemed  to  me  that  each  theory  of  which  I  have 
read  was  effectually  disposed  of  by  the  discus- 
sion which  it  aroused.  However  that  may  be, 
there  is  certainly  room  for  another  suggestion, 
and  this  I  proceed  to  offer. 

Reynolds  *  arranged  an  experiment  in  which 
a  liquid  was  made  to  flow  over  another  denser 
liquid,  the  two  being  immiscible.  Below  a  cer- 

i  Roy.  Soc.  London  Philos.  Trans.,  vol.  174,  pp.  943-944, 1883. 


tain  velocity  the  surface  of  contact  was  smooth, 
but  above  the  critical  velocity  the  surface  was 
occupied  by  a  system  of  equal  waves,  which 
moved  in  the  direction  of  flow  but  more  slowly. 
With  miscible  liquids,  or  with  two  bodies  of 
identical  liquid,  the  waves  are  replaced  by  vor- 
tices. Under  some  conditions  the  vortices  are 
as  regularly  spaced  as  the  waves,  but  usually 
they  are  less  regular,  and  various  complications 
arise.  The  development  of  such  vortices  may 
readily  be  watched  on  a  river  surface  wherever 
adjacent  parts  move  with  quite  different  veloci- 
ties or  move  in  opposite  directions.  Such  vor- 
tices have  vertical  axes,  and  their  direction  of 
rotation  is  determined  by  the  differential  mo- 
tion of  the  adjacent  currents.  If  we  conceive 
the  water  of  a  vortex  as  a  body  between  parallel 
and  opposed  currents,  then  its  direction  of  rota- 
tion is  due  to  a  mechanical  couple  contributed 
by  the  currents. 

Transferring   attention   to   the   longitudinal 
vertical  section  of  R  stream,  we  find  ft  couple  of 


FIGURE  S3.— Ideal  longitudinal  section  of  a  stream,  illustrating  hypothe- 
sis to  account  for  the  subsurface  position  of  the  level  of  maximum 
velocity. 

which  one  element  is  the  general  forward  move- 
ment of  the  current  and  the  other  is  the  bed 
resistance.  These  tend  to  produce  and  main- 
tain vortices  with  horizontal  axes  transverse 
to  the  channel  and  with  forward  rotation — that 
is,  with  the  rotation  of  a  wheel  rolling  forward  in 
the  direction  of  flow.  To  visualize  these  fea- 
t  ures,  figure  83  gives  an  ideal  section  of  a  stream, 
with  flow  from  left  to  right,  and  the  ovals  A,  B,  C 
represent  a  system  of  forward-rolling  vortices. 
The  arrows  within  an  oval  show  direction  of 
rotation,  and  it  is  important  to  recognize  that 
the  motions  they  indicate  are  referred  to  the 
center  of  the  vortex,  or  to  the  vortex  as  a 
whole,  and  not  to  the  fixed  bed  of  the  channel. 
With  reference  to  the  bed  all  parts  of  the  vortex 
are  moving  toward  the  right,  the  lower  part 
merely  moving  slower  than  the  upper. 

The  tendency  of  vortices  toward  circular 
forms  leaves  certain  tracts  of  the  section  unoc- 
cupied by  the  system  of  vortices.  Consider  the 
tract  D,  bounded  below  by  the  bed  and  above 


PROBLEMS  ASSOCIATED   WITH   KHYTHM. 


249 


by  parts  of  vortices  A  and  B,  and  give  attention 
to  the  motions  by  which  the  water  in  the  tract 
is  surrounded  and  influenced,  taking  care  to  re- 
fer each  motion  to  the  middle  of  the  tract  itself. 
Thus  referred,  the  motion  of  the  adjacent  part 
of  the  rear  vortex  A  is  to  the  left  and  down- 
ward, that  of  the  forward  vortex  B  is  to  the 
left  and  upward,  and  that  of  the  bed  is  to  the 
left.  The  motions  are  indicated  by  arrows. 
The  influences  of  the  vortices  tend  to  make  the 
water  about  D  rotate  backward,  while  the  influ- 
ence of  the  bed  tends  to  make  it  rotate  forward. 
The  result  is  not  a  priori  evident,  but  may  be 
assumed  to  be  something  different  from  simple 
rotation.  Its  possibilities  will  again  be  re- 
ferred to. 

Now  consider  the  tract  E,  bounded  below  by 
parts  of  A  and  B  and  above  by  the  water 
surface,  and  give  attention  to  the  motions  by 
which  its  water  is  affected.  Motions  being 
referred  as  before  to  the  tract  itself,  that  of 
vortex  A  is  to  the  right  and  downward,  and 
that  of  vortex  B  is  to  the  right  and  upward. 
Above  is  the  motion  of  the  air,  which,  in  the 
absence  of  wind,  is  to  the  left,  as  indicated  by 
an  arrow.  The  three  influences  to  which  the 
tract  of  water  is  subject  all  tend  to  give  a 
backward  rotation,  as  indicated  in  a  similar 
position  at  F.  The  vortex  F  is  secondary  to 
the  A,  B,  C system  and  rotates  in  the  opposite 
direction.  If  its  motions  be  referred  to  the 
fixed  stream  bed,  it  is  evident  that  the  water 
in  its  upper  part  moves  in  the  direction  of  the 
current  less  rapidly  than  the  water  in  its  lower 
part.  The  existence  of  such  a  vortex  therefore 
tends  to  reduce  the  average  velocity  at  the 
surface  of  the  current  and  increase  it  at  some 
lower  level. 

Abandoning  now  the  specific  and  ideal  case, 
we  may  state  the  hypothesis  in  general  terms. 
Among  the  important  causes  of  vortical  motion 
in  a  river  or  other  stream  is  the  mechanical 
couple  occasioned  by  the  general  forward 
motion  of  the  water  in  conjunction  with  the 
resistance  of  the  bed.  This  tends  to  form 
vortices  with  horizontal  axes  and  forward  roll; 
and  the  tendency  is  probably  strongest  in  the 
lower  part  of  the  stream.  In  a  space  adjacent 
to  two  forward-rolling  vortices  there  exists  a 
tendency  toward  the  development  of  a  second- 
ary, backward-rolling  vortex,  but  this  tendency 
is  apt  to  be  nullified  by  other  and  adverse 
influences  except  in  the  upper  part  of  the 


stream.  The  free  surface  of  the  water  does 
not  oppose  the  development  of  such  reversed 
vortices.  Wherever  reversed  vortices  abound 
the  velocity  at  the  surface  (averaged  with 
respect  to  time)  is  less  than  at  some  level  below 
the  surface. 

The  hypothesis  as  stated  has  no  stationary 
element,  but  the  phenomena  of  incipient  dunes 
show  that  in  certain  cases  repetitive  motions 
are  associated  with  stationary  space  divisions. 
A  supplementary  suggestion  assigns  these 
repetitive  motions,  or  their  initial  phases,  to 
the  triangular  space  D  in  the  ideal  diagram, 
figure  83.  The  forces  tending  toward  rotation 
in  that  space  are  antagonistic;  and  the  sugges- 
tion is  (1)  that  they  produce  some  sort  of 
alternating  movement  with  regular  periodicity, 
and  (2)  that  the  time  interval  of  this  move- 
ment, in  combination  with  the  forward  move- 
ment of  the  major  vortices  A,  B,  C,  yields  a 
stationary  space  interval. 

An  investigation  based  on  this  line  of  sugges- 
tion and  designed  to  test  it  should  lead  also  to 
an  explanation  of  the  observed  changes  in  the 
method  by  which  traction  is  accomplished. 
The  dune  method  is  associated  with  a  depth 
which  is  large  in  relation  to  the  mean  velocity 
and  with  a  moderate  bed  resistance.  It  is 
replaced  by  other  methods  in  consequence  of 
(1)  a  reduction  of  depth,  or  (2)  an  increase  of 
velocity,  or  (3)  an  increase  of  resistance. 
Reduction  of  depth  diminishes  the  space  for 
development  of  the  hypothetic  vortices.  In- 
crease of  velocity  or  of  resistance  tends  to 
enlarge  their  pattern.  In  either  case  the 
coercion  of  the  water  surface  restricts  the 
freedom  of  vortical  movements  and  imposes 
conditions  tending  to  modify  their  system. 

THE  MOVING  FIELD. 

The  competent  investigator  is  resourceful  in 
the  creation  of  new  apparatus  and  methods  as 
the  need  for  them  arises.  While  fully  conscious 
of  this  fact,  and  of  the  further  fact  that  no 
device  is  of  assured  value  till  it  has  been  tried, 
I  yet  can  not  forbear  to  mention,  for  the  benefit 
of  others,  a  method  which  the  Berkeley  experi- 
ence leads  me  to  think  valuable  for  the  study  of 
the  internal  details  of  a  current  of  water. 

In  a  current  limited  by  the  sides  of  a  narrow, 
straight  trough  transverse  movements  are 
largely  suppressed,  so  that  most  of  the  action 
can  be  learned  from  observation  of  what  takes 


250 


TRANSPORTATION    OF   DEBRIS   BY   RUNNING   WATER. 


place  ill  vertical  planes  parallel  to  the  axis. 
The  movements  in  a  vertical  plane  may  be 
exhibited  by  giving  to  the  trough  a  glass  side 
and  by  giving  to  the  water  in  that  plane  ex- 
clusive illumination.  When  one  of  our  labora- 
tory currents,  being  illuminated  from  above,  was 
viewed  from  the  side,  small  particles  in  suspen- 
sion were  seen  to  be  conspicuous,  so  conspicu- 
ous, in  fact,  as  to  resemble  the  motes  in  a  sun- 
beam. Some  of  the  particles  were  shreds  of 
wood  fiber  worn  from  the  trough,  and  these 
could  be  more  easily  followed  by  the  eye 
because  of  their  distinctive  forms.  Impressed 
by  this  phenomenon,  I  am  confident  that  water 
movements  in  a  vertical  plane  can  be  effec- 
tively revealed  by  giving  to  the  water  a  suitable 
amount  of  suitable  suspended  material  and  by 
giving  to  the  selected  plane  a  brilliant  and  ex- 
clusive illumination. 

With  the  aid  of  such  simple  arrangements 
much  may  be  seen,  but  measurement  will  be 
difficult;  and  vortical  movements  may  not  be 
easily  discriminated  from  those  which  are 
merely  sinuous.  It  is  believed,  however,  that 
both  these  results  may  be  achieved  by  aid  of 
the  moving  field.  In  a  very  simple  form  this 
device  was  employed  by  us  in  the  study  of 
processes  of  traction  (p.  27),  and  despite  the 
crudity  of  the  apparatus  it  was  found  to  be 
highly  efficient.  An  experiment  trough  having 
in  its  side  a  glass  panel,  A  A  in  figure  4,  bore  a 
sliding  screen,  B,  in  which  was  an  opening,  C. 
Moving  debris  was  watched  through  the  open- 
ing at  the  same  time  that  the  screen  and  open- 
ing were  moved  in  the  direction  of  the  current. 
To  the  field  of  view  was  thus  given  a  hori- 
zontal motion,  and  that  motion  was,  in  effect, 
subtracted  from  the  motions  of  the  objects 
observed.  The  apparent  motions  were  motions 
in  relation  to  the  moving  field,  and  not  to  the 
fixed  trough.  No  provision  was  made  for  de- 
termining the  velocity  of  the  field,  nor  for 
measuring  the  apparent  motions  of  the  objects 
viewed;  but  even  without  these  devices  for 
quantitative  work,  the  possibilities  of  the 
method  of  observation  were  sufficiently  evi- 
dent. With  the  necessary  supplementary 
devices,  the  moving  field  promises  to  measure 
the  horizontal  and  vertical  components  of 
motion  hi  any  part  of  the  vertical  section,  and 


it  thus  makes  possible  the  complete  delineation 
of  velocities  and  directions  of  details  of  current, 
so  far  as  those  details  may  be  exhibited  in 
longitudinal  vertical  sections.  By  giving  to 
the  field  a  suitable  velocity  it  should  be  possi- 
ble to  see  a  traveling  vortex  which  rotates  in  a 
vertical  plane,  just  as  vortices  of  horizontal 
rotation  are  seen  on  the  surface  of  a  stream. 

A  notable  defect  in  the  Berkeley  arrangement 
was  the  requirement  that  the  observer  move 
his  head  in  unison  with  the  moving  field.  This 
interfered  with  steadiness  and  also  limited  nar- 
rowly the  space  covered  by  an  observation.  It 
could  be  remedied  by  substituting  for  the  slide 
a  car  which  should  carry  both  observer  and 
peephole  at  a  determined  rate. 

Another  suggested  arrangement  places  the 
eye  of  the  observer  at  a  fixed  telescope  and 


A 


FIGURE  84.— Diagrammatic  plan  of  suggested  moving-field  apparatus. 

moves  the  field  by  means  of  a  rotating  mirror. 
This  is  illustrated  by  figure  84,  where  the 
trough  T  is  shown  in  plan.  The  telescope  E 
views  the  trough  through  the  mirror,  which  is 
pivoted  on  a  vertical  axis  at  M.  As  the  plane 
of  the  mirror  rotates  from  a  to  i  the  field  com- 
manded by  the  telescope  moves  from  A  to  B. 
It  is  evident  that  if  the  angular  velocity  of  the 
mirror  is  constant  the  linear  motion  of  the  field 
along  the  trough  will  be  relatively  fast  at  A 
and  B  and  relatively  slow  at  0.  The  error  thus 
arising  may  be  avoided  by  some  device  of  the 
nature  of  linkage.  It  will  be  corrected  with 
sufficient  approximation  if  the  axis  of  the  mir- 
ror be  controlled  by  a  rigidly  attached  arm 
MD,  which  is  in  turn  controlled  by  an  arm 
about  one-third  as  long,  the  two  having  sliding 
contact  at  D  and  the  short  arm  revolving  uni- 
formly about  a  vertical  axis  at  F. 


APPENDIX  A.— THE  PITOT-DARCY  GAGE. 


SCOPE    OF    APPENDIX. 

Many  measurements  of  velocity  were  made 
with  the  Pitot-Darcy  gage,  but  as  only  a  few 
have  been  finally  utilized  in  the  preparation 
of  the  report,  a  discussion  of  the  instrument 
seemed  not  appropriate  to  the  main  text. 
Certain  phases  of  our  experience,  however, 
are  thought  worthy  of  record  because  they 
have  practical  bearing  on  the  utility  and  the 
use  of  such  gages,  and  these  are  the  subject  of 
the  appendix. 

FORM    OF    INSTRUMENT. 

Darcy  developed  the  Pitot  tube  by  adding 
a  second  tube,  differently  related  to  the  cur- 
rent, and  by  connecting  the  two  above  with  a 
chamber  from  which  the  air  was  partly  ex- 
hausted. The  water  columns  in  the  two  tubes 
were  thus  lifted  from  the  vicinity  of  the  water 
surface  to  a  convenient  position,  where  their 
difference  in  height  could  readily  be  measured. 
In  the  gages  constructed  for  our  use  the 
aperture  of  one  tube  was  directed  upstream 
and  that  of  the  other  downstream.  The 
tubes  were  borings  in  a  single  piece  of  brass, 
which  was  shaped  on  the  outside  in  smooth 
contours,  designed  to  interfere  the  least  possible 
with  the  movement  of  the  water.  The  form 
first  given  was  afterward  modified,  and  figure 
85  shows  the  third  and  last  design,  with  which 
most  of  the  work  was  done.  The  openings 
had  a  diameter  of  0.1  inch.  At  the  opposite 
or  upper  end  of  the  brass  piece  were  stopcocks, 
and  above  these  connection  was  made  with 
rubber  tubes,  which  led  to  the  complementary 
part  of  the  apparatus,  where  the  difference  in 
height  of  the  two  water  columns  was  observed. 
It  is  convenient  to  call  tiie  member  exposed  to 
the  current  the  receiver,  and  the  complemen- 
tary member  the  comparator. 

In  the  comparator  were  two  glass  tubes, 
straight  and  parallel,  with  internal  diameters 
of  about  0.8  inch.  At  the  top  they  were 
connected  by  an  arch,  and  at  the  summit  of  the 
arch  was  a  branch  tube,  with  a  pet  cock,  used 
in  regulating  the  amounts  of  air  and  water. 


At  the  bottom  they  communicated  with  the 
rubber  tubes  through  a  brass  piece,  in  which 
were  two  stopcocks,  connected  by  gearing  so 
as  to  open  and  close  together.  These  parts 
were  mounted  on  a  board  which  also  carried  a 
scale  of  inches  and  decimals.  A  sliding  index 
was  arranged  so  that  it  could  be  set  by  the 
meniscus  of  a  water  column  and  its  position 
then  read  on  the  scale;  and  there  was  a  fixed 
mirror  behind  the  tubes  to  aid  in  avoiding 
error  from  parallax.  The  board  was  sup- 
ported in  an  inclined  position,  the  slope  given 
to  the  tubes  and  scale  being  that  of  2£  hori- 
zontal to  1  vertical.  This  had  the  effect  of 


0.1 


o.a  root 


FIGURE  85.— Longitudinal  section  of  lower  end  of  receiver  of  Pitot-Darcy 
gage  No.  3,  with  transverse  sections  at  three  points. 

making  the  movements  of  the  columns  2.69 
times  as  great  for  the  same  change  of  pressure 
as  they  would  be  if  the  tubes  were  vertical. 

In  preparing  for  observation,  the  internal  air 
pressure  was  so  adjusted  that  the  columns 
stood  near  the  middle  of  the  scale.  The 
receiver  was  then  held,  by  a  suitable  frame,  in 
the  selected  part  of  the  current,  and  the  stop- 
cocks were  opened.  In  the  glass  tube  con- 
nected with  the  receiver  opening  facing  up- 
stream the  column  rose;  in  the  other  tube  it 
fell.  When  the  full  effect  of  the  current  had 
been  realized,  the  stopcocks  at  the  bottom  of 

251 


252 


TRANSPORTATION    OF   DEBRIS   BY   RUNNING    WATER. 


the  comparator  were  closed,  and  the  heights 
of  the  columns  were  then  read. 

The  use  of  the  gage  to  measure  velocities 
close  to  the  bed  of  debris  proved  impracticable 
because  the  presence  of  the  receiver  modified 
the  movement  of  the  water  and  thereby  modi- 
fied the  shape  of  the  bed.  (See  pp.  26, 155.) 
This  effect  could  have  been  reduced  by  using  a 
different  form  of  receiver.  Darcy  and  Bazin  * 
bent  the  tubes  at  the  bottom  in  such  a  way 
that  one  or  both  openings  met  the  water  at 
some  distance  upstream  from  the  vertical  part 
of  the  tubes,  and  it  is  probable  that  the  adop- 
tion of  their  design  would  have  diminished  the 
difficulty,  although  it  could  not  have  removed 
it.  With  such  a  design,  however,  the  practi- 
cable forms  for  the  second  opening  relate  it  to 
the  piezometer,  and  the  advantage  of  the  down- 
stream opening  is  lost. 

That  part  of  the  design  of  the  comparator 
which  consists  in  the  inclination  of  tubes  and 
scale  is  not  to  be  recommended.  It  refines  by 
magnifying  the  reading,  but  it  introduces  pos- 
sibilities of  error  in  other  ways.  If  the  tubes 
are  not  straight  or  are  not  equally  inclined,  an 
error  is  occasioned  which  does  not  enter  if  they 
and  the  scale  are  vertical.  Evidence  of  such 
error  was  found  in  the  fact  that  the  still-water 
or  zero-velocity  readings  of  the  two  columns 
were  not  always  identical,  but  no  ready  means 
of  correction  was  discovered. 

Another  source  of  error  was  detected  in  ine- 
qualities of  sectional  area  of  the  glass  tubes  of 
the  comparator.  To  show  the  nature  of  this 
error,  let  us  assume  that  the  pressure  of  the 
current  at  the  upstream  opening  of  the  receiver 
is  exactly  equal  to  the  negative  pressure,  or 
suction,  at  the  downstream  opening.  If  the 
glass  tubes  are  of  uniform  and  equal  bore,  one 
column  moves  upward  just  as  much  as  the  other 
moves  downward,  and  the  volume  of  air  above 
the  columns  is  unchanged.  Now,  assume  that 
the  tube  containing  the  rising  column  has  the 
greater  diameter.  It  is  evident  that  equal 
movement  of  the  two  columns  will  displace 
more  air  in  the  one  tube  than  it  will  provide 
space  for  in  the  other,  and  the  pressure  of  the 
confined  air  will  be  thereby  increased.  The 
effect  of  the  increased  pressure  will  be  to  lower 

i  Bazin,  F.  A.,  Recherches  experimentales  sur  1'ecoulement  de  1'eau 
dans  les  canaux  decouverts:  Acad.  sci.  Paris  Mem.  math,  et  phys., 
vol.  19,  p.  49,  PI.  IV,  18fi5.  This  memoir  was  published  also  as  part  of 
"Recherches  hydrauliques,"  by  H.  Darcy  and  F.  A.  Bazin. 


both  columns.  There  is  also  a  secondary  effect 
of  small  amount  connected  with  the  fact  that 
the  pressure  of  the  confined  air  plus  the  head 
of  water  between  the  tops  of  the  columns  and 
the  surface  of  the  stream,  on  the  one  hand,  and 
the  atmospheric  pressure,  on  the  other,  are  in 
equilibrium,  but  into  this  we  need  not  here 
enter. 

As  a  means  for  the  discussion  of  these  errors 
the  tubes  weie  calibrated,  by  Prof.  J.  N.  Le 
Conte,  in  the  following  manner:  The  tubes 
being  closed  at  the  bottom,  a  weighed  quantity 
of  water  was  introduced  into  one  and  the  height 
of  its  column  was  read.  By  lepeated  additions 
of  water  and  repeated  readings  the  volumes  of 
divisions  of  each  tube  were  thus  measured,  the 
divisions  being  approximately  1.5  inches  in 
length.  In  similar  manner  the  volume  was 
measured  of  the  space  above  the  straight  tubes 
to  the  pet  cock.  The  average  sectional  area  of 
one  tube  was  found  to  be  2.5  per  cent  greater 
than  that  of  the  other.  The  sectional  area  in 
the  larger  tube  was  found  to  vary  through  a 
range  of  at  least  2.6  per  cent,  and  the  range  for 
the  smaller  tube  was  4.5  per  cent. 

A  table  of  corrections  to  readings  was 
computed  from  the  data  of  calibration,  and 
this  table  was  practically  applied.  In  com- 
paring the  rise  of  one  column  with  the  asso- 
ciated fall  of  the  other  the  greatest  correction 
applied  amounted  to  a  little  less  than  1  per 
cent.  In  the  determination  of  velocity  the 
largest  correction  applicable  for  this  reason  was 
about  0.3  per  cent.  These  corrections,  how- 
ever, pertain  only  to  the  discussions  of  the  in- 
strument in  the  following  pages.  In  the 
ordinary  work  of  the  gage  they  were  not  ap- 
plied, because  it  was  found  that  the  errors  of 
this  class  were  practically  eliminated  when  the 
same  methods  were  employed  in  the  prepara- 
tion and  in  the  use  of  rating  formulas.  The 
matter  is  mentioned  here  chiefly  because  errors 
from  unequal  tube  caliber,  which  may  some- 
times prove  important,  appear  not  to  have 
been  allowed  for  in  the  discussions  of  Pitot- 
Darcy  gages. 

BATING    METHODS. 

The  first  and  second  gages  were  rated  by  the 
method  of  floats;  the  third  was  twice  rated  by 
the  running-water  method  and  several  times  by 
the  still-water  method.  The  floats  used  weie 


THE   PITOT-DARCY   GAGE. 


253 


vertical  cylinders  of  wood  so  adjusted  that  the 
submerged  depth  was  twice  the  distance  below 
the  water  surface  of  the  apertures  of  the  re- 
ceiver. Some  of  the  still-water  ratings  were 
made  at  the  Geological  Survey's  rating  station 
at  Los  Angeles,  where  a  car  lunning  on  a  track 
at  the  side  of  a  reservoir  drew  the  receiver 
through  the  water  of  the  reservoir.  The  others 
were  ma  do  la  ter  in  the  long  trough  of  the  Berkeley 
laboratory,  the  car  in  this  case  running  above 
the  water.  In  the  application  of  the  running- 
water  method  a  measured  discharge  was  passed 
through  a  rectangular  trough  and  a  survey  of 
velocities  throughout  a  cross  section  was  made 
by  means  of  the  gage.  By  using  in  this  survey 
the  rating  formula  obtained  by  the  still-water 
method,  and  then  comparing  the  mean  velocity 
thus  computed  with  that  computed  from  the 
discharge  and  sectional  area,  a  correction  is 
obtained  which  may  be  applied  to  the  still- 
water  rating  formula.  Only  a  single  compari- 
son of  this  sort  is  practically  available  in  con- 
nection with  our  instruments,  and  the  terms  of 
this  are: 

Ft./sec. 

Mean  velocity  by  discharge  and  area 1. 98 

Mean  velocity  by  gage  (with  still-water  rating). 2. 08±0. 07 

The  resulting  correction  to  the  still-water 
ratings  is  — 5  per  cent,  but  this  determination 
has  small  value  because  of  the  large  probable 
error  of  one  of  the  compared  determinations. 

The  fact  that  the  apparent  correction  is 
small  is  in  accord  with  a  property  of  the  gage 
independently  observed.  By  reason  of  the 
sinuosity  of  flow  lines  in  a  stream,  the  direc- 
tions of  motion  are  not  parallel  to  the  axis  of  a 
straight  channel.  Therefore,  a  current  meter 
wliich  records  the  velocity  in  the  direction  of 
flow  instead  of  the  component  of  velocity  par- 
allel to  the  channel  axis  yields  an  overestimate 
of  mean  velocity.  Some  Pitot-Darcy  gages 
have  been  found  to  overestimate  velocity  when 
placed  obliquely  to  the  direction  of  flow,  and 
for  such  the  correction  would  be  large.  It  was 
found,  however,  that  the  Berkeley  gage  when 
placed  somewhat  obliquely  to  the  current  gave 
a  lower  reading  than  when  facing  it  squarely, 
and  through  this  property  it  tended  auto- 
matically to  correct  its  readings  for  obliquity 
of  current.  If  the  correction  were  perfect,  the 
still-water  rating  and  running-water  rating 
should  be  the  same. 


RATING    FORMULA. 

In  the  still-water  ratings  velocities  inde- 
pendently determined  were  compared  with 
resulting  changes  in  the  water  columns  of  the 
comparator.  Starting  from  the  same  level, 
one  column  rose  with  increase  of  speed  and  the 
other  fell.  Except  for  the  influence  of  modi- 
fying conditions  the  changes  should  be  equal, 
the  positive  velocity  head  being  of  the  same 
amount  as  the  negative  velocity  head.  To  test 
this  point  various  sets  of  observations  were 
plotted  on  section  paper,  the  readings  of  the 
rising  column  being  taken  as  abscissas  and  the 
readings  of  the  f  ailing  column  as  ordinates.  In 
most  cases  the  plotted  points  fell  well  into  line, 
and  there  was  no  question  that  the  line  repre- 
sented by  them  was  straight.  That  is,  the 
true  ratio  of  the  negative  pressure  at  the  down- 
stream opening  of  the  receiver  to  the  positive 
pressure  at  the  upstream  opening  was  constant, 
under  the  conditions  of  this  particular  series  of 
trials.  The  value  of  the  ratio  was  found  to  vary 
with  conditions,  and  the  several  values  found 
are  so  near  unity  as  to  confirm  the  theoretic 
belief  that  unity  is  the  normal  value. 

In  considering  the  variations  of  value  it  is 
first  to  be  noted  that  the  gage  with  wliich  the 
observations  were  made,  No.  3,  being  sym- 
metric, could  have  either  opening  turned  up- 
stream, and  it  was  in  fact  used  both  ways,  with 
record  of  its  position.  But  its  symmetry  was 
only  approximate  and  therefore  the  two  posi- 
tions gave  different  results.  Used  in  one  way 
it  will  be  called  No.  3a,  and  in  the  other  way 
No.  3b.  It  also  happened  that  between  the 
date  of  the  Los  Angeles  ratings  and  that  of  the 
Berkeley  ratings  the  receiver  was  accidently 
marred  at  one  of  its  openings,  and  though  its 
form  was  afterward  restored  as  nearly .  as 
possible,  some  difference  remained  which  af- 
fected its  constants.  The  values  of  the  pressure 
ratio  are  accordingly  arranged  in  four  groups 
in  Table  81.  Two  of  these  groups  also  are 
subdivided  with  reference  to  the  position  of 
the  receiver  in  relation  to  the  perimeter  of  the 
current. 

The  ratio  was  notably  larger  after  the  acci- 
dent than  before,  and  the  change  was  greater 
for  3a  than  for  3b.  The  greatest  value  of  the 
ratio  was  given  by  trials  in  which  the  receiver 
ran  close  to  the  side  of  the  reservoir,  which  in 


254 


that  case  was  a  plank  trough.  The  interpreta- 
tion  of  these  results  will  be  considered  in 
another  connection. 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 

Substituting  in  (118),  we  have 

iff- fl.- 82.28  fcF*. 


TABLE  81. — Ratio  of  the  suction  at  one  opening  of  the  Pitot- 
Darcy  gage  to  the  pressure  at  the  other. 


Place  of  rating. 

Conditions. 

Ratio,  suction  to  pressure. 

Gage  3a. 

Gage  3b. 

Los  Angeles  .... 

Deep  water 

0.993±0.015 
1.038±  .007 

1.005±0.005 
1.010±  .002 
1.016±  .003 
1.014±  .005 

Do  

Do  
Do  

Shallow,  near  surface.  . 

Shallow,  near  side  

1.052±   .004 

Let  A.  represent  the  vertical  space  through 
which  the  column  of  water  is  raised  by  pressure 
from  velocity  V  on  the  upstream  opening  of  the 
receiver,  and  At  the  simultaneous  depression  of 
the  column  connected  with  the  downstream 
opening.  Then,  each  being  assumed  to  equal 
the  velocity  head, 


V2 


-(116) 
-(117) 


In  practice  the  full  velocity  head  is  not  realized 
in  instruments  of  the  Pitot-Darcy  type,  and  the 

r) 

coefficient  determined  by  rating  is  less  than  —  . 

y 

It  is  a  common  experience  also  in  practical 
application  of  hydraulic  formulas  to  find  that 
qualification  is  advisable  in  other  respects.  I 
therefore  substituted  tentatively  for  (117)  the 
formula 


^JcV* 


(118) 


and  sought  empiric  values  of  u  and  &.  The 
readings  of  the  comparator  corresponding  to  A, 
and  A!  may  bo  called  H  and  11^.  The  zero  of 
the  comparator  scale  being  at  its  lower  end, 
the  difference  between  the  readings  corres- 
ponds to  the  sum  of  the  spaces  A.  and  \.  The 
readings  are  in  inches,  while  the  unit  used  for  A, 
A1;  and  g  is  the  foot.  Moreover,  from  the  incli- 
nation of  the  comparator,  the  space  between 
the  two  columns,  as  read  on  the  scale,  is  2.09 
times  the  vertical  space  Ji  +  Ji^  Therefore,  the 
product  of  12  by  2.69  being  32.28, 

//-//,  =32.28  (A  +  &,)_„.     ..(119) 


or,  making  A'=  32.38  Jc, 

11-11,=  KVU (121) 

The  observed  quantities  being  //,  7/1;  and  V, 
it  was  possible  to  plot  on  logarithmic  section 
paper  any  series  of  values  of  H—  Ht  in  relation 
to  the  associated  values  of  V,  and  thus  compute 
graphically  the  corresponding  values  of  u  and 
K.  The  values  were  computed  for  all  series  of 
observations  represented  in  Table  81,  and  they 
are  given  in  Table  82.  The  mean  of  the  seven 
values  of  u  is  2.00,  but  their  range  is  notable. 
The  deviations  from  the  normal  may  be  ascribed ' 
in  part  to  accidental  errors.  In  the  case  of  the 
third  value,  1.90,  and  of  the  seventh,  1.94,  the 
plotted  positions  are  so  scattered  as  to  admit 
of  considerable  latitude  in  the  drawing  of  the 
equation  lines,  but  the  control  is  much  stronger 
for  the  values  2.09  and  2.12,  and  these  could 
not  be  greatly  reduced  without  violence  to  the 
facts  of  observation.  It  seems  clear  that  the 
exponent  is  not  whouy  free  from  the  influence 
of  special  conditions. 

TABLE  82.—  Values  of  K  and  u  in  //-//I=A'T'«. 


Place  of  rating. 

Conditions. 

Gage. 

X 

u 

Los  Angeles... 

Deep  wator 

3a 

0  70 

Do  .  

.do.... 

3b 

Berkeley. 

Do... 

do  . 

3b 

Do  

3b 

70 

Do... 

3b 

Do.. 

For  the  practical  purpose  of  rating  the  in- 
strument, however,  there  is  no  advantage  in 
departing  from  the  normal  exponent,  and  that 
was  employed  in  the  preparation  of  rating 
tables.  The  formula  used  for  inferring  veloci- 
ties from  readings  is 


.(122) 


in  which 


The  values  of  A}  graphically  computed,  are 
given  in  Table  83,  and  these  values  were  used 
in  the  computations  of  velocities. 


THE   PITOT-DARCY   GAGE. 


255 


TABLE  83. — -Values  of  A  in  V=A*^h  —  JJt  and  values  of  gage  efficiency. 


]  'lace  of  rating. 

Conditions. 

Values  of  A. 

Efficiency  of 
gage. 

Gage  3a. 

(iageSb. 

3a. 

3b. 

Deep  water                    

1.22±0.02 
1.29±   .04 

1.2C±0.01 
1.30±  .02 
1.24±  .01 

0.67 
.60 

0.63 
.59 
.66 

.53 

Do                            .     .   -. 

Shallow,  near  bed             

Do 

1  37±   .03 

Do 

1  26±     03 

63 

An  inspection  of  the  values  of  A  with  due 
attention  to   their  probable  errors  serves   to 
show  that  their  differences  are  not  to  be  re- 
garded as  wholly  accidental,  but  must  be  as- 
cribed in  part  to  the  variation  of  the  instru- 
mental constant  with  conditions.     That   the 
nature  of  the  variations  may  be  appreciated, 
the  conditions  will  be  more  fully  described. 
As  already  stated,  gages  3a  and  3b  are  the  same 
symmetric  instrument,  but  with  opposite  faces 
turned  toward  the  current,  while  the  instru- 
ment was  modified  to  some  minute  extent  by 
an  injury  and  repair  occurring  between  the  Los 
Angeles    and    Berkeley    ratings.     Practically 
there  were  four  gages,  but  so  far  as  the  mech- 
anician could  make  them  the  four  gages  were 
identical  in  form.     The  reservoir  at  the  Los 
Angeles  rating  station  was  broad  and  deep. 
The  course  followed  by  the  receiver  of  the  gage 
was  several  feet  from  the  bank,  at  least  6  inches 
below  the  surface,  and  at  least  1  foot  above  the 
bottom.     The  plank  reservoir  used  in  Berkeley 
was  1.96  feet  wide,  and  the  depth  of  water  was 
0.44  foot.     In  the  ratings  tabulated  as  at "  mid- 
depth"  the  opening  of  the  receiver  was  central 
in  the  cross  section  of  the  water.     In  the  "  near 
bottom"  rating  the  center  of  the  opening  (its 
diameter  being  0.01  foot)  was  0.02  foot  from 
the  bottom;  in  the  "near  surface"  rating  the 
center  was  0.03  foot  below  the  surface  of  the 
water;  and  in  the  "near  side"  rating  the  cen- 
ter was  0.25  foot  above  the  bottom  and  varied 
from  0.01  to  0.11  foot  in  its  distance  from  the 
side,  the  course  and  the  trough  side  not  being 
quite  parallel.     The  several  relations  of  the 
opening  to  the  water  section  are  shown  in  fig- 
ure 86. 

There  is  nothing  novel  in  the  variation  of  the 
instrumental  constant  with  minute  differences 
in  the  shape  of  the  receiver.  This  has  been 
observed  by  all  critical  users  of  such  instru- 
ments, and  the  custom  is  well  established  of 


giving  a  separate  rating  to  each  receiver.  •  So 
far  as  I  am  aware  the  variation  of  the  constant 
with  the  relation  of  the  receiver  to  the  walls  of 
the  conduit  and  to  the  water  surface  has  not 
previously  been  recorded,  and  this  is  a  matter 
of  considerable  moment,  for  various  elaborate 


FIGURE  8ti. — Cross  section  of  prism  of  water  in  trough,  showing  positions 
given  to  gage  opening  in  various  ratings. 

discussions  of  the  distribution  of  velocities 
within  a  cross  section  have  assumed  the  con- 
stancy of  the  instrumental  constant  for  all  posi- 
tions. As  clearly  indicated  by  the  tabulated 
results,  the  constant  A  increases  when  the  free 
surface  of  the  water  is  approached  and  dimin- 
ishes when  a  rigid  wall  of  the  conduit  is 
approached. 

In  the  application  of  these  results  (and  of 
cognate  results  with  which  it  was  thought  best 
not  to  burden  these  pages)  to  our  velocity  de- 
terminations a  graphic  table  was  constructed 
to  show  the  relation  of  the  instrumental  con- 
stant to  the  position  of  receiver  in  the  conduit; 
and  this  table  (fig.  87)  supplied  the  constant 
to  be  used  with  each  individual  observation. 
This  procedure  had  an  important  influence  on 
the  interpretation  of  the  running-water  rating, 
making  the  apparent  correction  for  that  rating 
less  than  it  would  be  if  the  mid-depth  constant 
were  used  exclusively.  Its  influence  on  the 
vertical  velocity  curves  (figs.  74  to  82)  was  to 
give  them  less  curvature  near  the  water  surface, 
greater  curvature  in  approaching  the  channel 
bed,  and  (sometimes)  a  somewhat  lower  level  of 
maximum  velocity. 

2o 

In  equation  (117)  -^  V2  equals  twice  the  theo- 
retic velocity  head.  If  in  equation  (118)  the 
exponent  u  be  replaced  by  its  mean  value  2, 


256 


TRANSPORTATION   OF   DEBRIS  BY   RUNNING   WATER. 


we  have  Jc  V2  as  an  expression  for  the  double 
head   actually   produced   by   the   gage.     The 

2 
ratio  of  Jc  to  —  -  measures  the  efficiency  of  the 

9 

instrument    in    realizing    the    theoretic   head. 

the 


ure  of  efficiency  is 


2      32.16   1       0.996 


32.28  A2  '  2g     32.28  A2  ~~  A2 

The  values  of  the  measure  have  been  com- 
puted and  are  given  in  Table  83.  The  deep- 
water  and  mid-depth  ratings  being  assumed  to 
represent  normal  conditions,  the  mean  of  the 
corresponding  efficiencies,  namely,  0.62,  may 
stand  in  a  general  way  for  the  fraction  of  the 
theoretic  head  which  is  realized  by  this  particu- 
lar type  of  the  Pitot-Darcy  gage. 

As  the  relation  between  velocity  and 
observed  head  is  that  between  agent  and  effect, 
it  is  evident  that  values  of  the  efficiency  ratio 
rather  than  those  of  the  constant  A  should  be 
compared  in  any  attempt  to  explain  the  phe- 
nomena of  variation.  Restating  from  this 
viewpoint  the  results  of  comparative  ratings, 
we  have:  The  response  of  the  head  to  changes 
of  velocity  is  lessened  when  the  receiver  of  the 
gage  is  brought  near  the  free  water  surface  and 
is  increased  when  the  receiver  is  brought  near 


a  rigid  part  of  the  stream's  perimeter.  It  may 
be  surmised  that  the  differences  in  head  are 
connected  with  the  facility  with  which  the 
flow  lines  of  the  water  are  diverted  in  passing 
around  the  instrument,  regarded  as  an  obstruc- 
tion. In  midcurrent  the  diversion  is  resisted 


Water  surface 


Bottom  or  side 


FIGURE  87.— Graphic  table  for  interpolating  values  of  A,  in  V= 
A  -jH—Hi,  for  observations  made  with  gage  3b  in  different  parts  of  a 
stream. 

by  the  inertia  of  surrounding  water.  Near 
the  surface  the  resistance  of  water  is  partly 
replaced  by  resistance  of  more  mobile  air. 
Near  the  conduit  the  resistance  of  water  is 
partly  replaced  by  the  resistance  of  an  immobile 
solid. 


APPENDIX   B.— THE  DISCHARGE-MEASURING  GATE  AND  ITS  RATING. 


THE    GATE. 

The  gate  for  measurement  of  discharge  is 
mentioned  at  page  20,  and  its  general  relation 
to  other  apparatus  is  shown  in  figure  2.  In  a 
section  on  the  measurement  of  discharge,  page 
22,  it  is  briefly  described,  and  the  history  of  its 
use  is  outlined.  Figure  88  is  designed  to  ex- 
hibit its  relations  more  fully  and  shows  the 
arrangement  of  the  general  apparatus  of  the 
laboratory  at  the  time  of  its  calibration. 


The  gate  controlled  an  opening  in  the  side  of 
a  vertical  wooden  shaft,  of  which  the  internal 
horizontal  section  measured  3  feet  by  0.5  foot. 
At  the  top  the  shaft  communicated  freely  with 
a  long,  narrow  tank,  to  which  water  was  con- 
tinuously delivered  by  a  pump.  The  surface 
level  of  water  in  the  tank  was  regulated  by 
means  of  an  overflow  weir  and  a  valve  above 
the  pump  and  was  observed  in  a  glass  tube  or 
gage,  AB,  in  figure  88,  outside  of  the  tank. 
This  tube  was  given  a  slope  of  1  in  10,  so  that 


Overhead  tank 


FIGURE  88. — Arrangement  of  apparatus  connected  with  the  rating  of  the  discharge-measuring  gate. 


the  movements  of  its  water  column  afforded  a 
magnified  indication  of  changes  in  the  water 
level  within  the  tank. 

Details  of  the  gate  are  represented  to  scale 
in  figure  89.  A  brass  plate,  PP,  attached  by 
screws  to  the  outer  face  of  the  shaft,  was 
pierced  by  an  opening  of  10.5  by  2  inches,  the 
longer  dimension  being  horizontal.  About 
three  sides  the  edges  of  the  opening  were 
beveled  outward,  leaving  a  45°  edge  at  the 

20921°— No.  86— 14 IT 


inner  face  of  the  plate.  A  sliding  plate,  G, 
rested  against  the  inner  face  of  the  fixed  plate. 
This  was  the  gate  proper,  its  function  being  to 
close  either  the  whole  or  a  definite  portion  of 
the  opening  and  thus  regulate  the  width  of  the 
issuing  water  jet.  It  overlapped  by  half  an 
inch  the  margins  of  the  opening,  above  and 
below,  and  was  guided  by  two  brass  pieces, 
which  appear  in  sections  6  and  c.  To  the  end 
adjacent  to  the  water  jet  was  given  a  chisel 

257 


258 


TRANSPORTATION   OF   DEBRIS   BY   RUNNING   WATER. 


edge.  The  brass  guides  and  the  adjacent  parts 
of  the  wooden  shaft  wall  were  shaped  to  a  45° 
bevel,  which,  if  produced,  would  reach  the  edge 
of  the  opening  in  the  fixed  plate. 

Movements  of  the  slide  were  controlled  from 
the  outside.  A  brass  rod,  0,  firmly  attached 
to  it  and  running  parallel  to  its  axis  (fig.  89J), 
rested  in  a  frame  at  the  left  of  the  opening 
(fig.  89«,  e),  where  its  motion  was  controlled  by 
rack  and  pinion,  the  rack  being  cut  on  the 
upper  side  of  the  rod.  At  the  left  of  the  frame 
and  pinion  the  rod  slid  along  a  brass  scale 
graduated  to  inches  and  tenths;  and  the  gate 
was  set  for  any  desired  width  of  aperture  by 
bringing  an  engraved  index  mark  on  the  rod 
opposite  the  proper  mark  on  the  scale. 


The  operation  of  the  gate  was  found  to  be 
satisfactory. 

The  head  under  which  the  jet  issued,  meas- 
ured from  the  middle  of  the  opening  to  the 
water  surface  in  the  upper  tank,  was  6.0  feet, 
plus  or  minus  a  small  fraction  observed  by 
means  of  the  gage  above  mentioned. 

PLAN    OF    EATING. 

The  method  of  rating  was  volumetric  and 
empiric.  The  gate  having  been  set  at  a  par- 
ticular graduation,  and  the  discharge  having 
been,  continued  until  the  rate  of  flow  through 
the  stilling  tank  and  experiment  trough  had 
become  steady,  the  outflow  of  the  trough  was 
diverted  for  a  measured  time  into  a  special 


p.- 

(a,) 

A 

O 
(d) 

Section  through  A-S 

p       /Vj=d\Nx 

t  ^     j4^^^Lup~^~^J^.  —          —s.  —  , 

Outer  face 
of  gate 

^B'                                  f 

ZS/////////X 


Section  through  C-D 
0  5  10 


Section  through  E-f 
15  20  ZS  Inches 


FIGURE  89.— Elevation  and  sections  of  gate  for  the  measurement  of  discharge. 


reservoir  where  its  volume  was  measured.  The 
special  reservoir  is  shown  in  figure  88  as  "  sump 
No.  2,"  and  the  diverting  apparatus  also  is 
indicated.  The  "diverting  trough"  was  piv- 
oted at  the  remote  end  and  could  be  turned 
quickly  by  hand.  In  one  position  it  received 
the  discharge  from  the  experiment  trough  and 
delivered  it  to  sump  No.  1,  containing  the 
supply  for  the  pump;  in  another  it  permitted 
the  water  to  fall  into  sump  No.  2,  arranged  for 
measurement.  The  work  of  rating  was  per- 
formed by  J.  A.  Burgess,  and  a  full  report  upon 
it  constituted  his  graduation  thesis  in  engi- 
neering at  the  University  of  California.  The 
details  of  apparatus  and  method  were  arranged 


by  him,  and  the  following  account  of  the  rating 
is  essentially  an  abridgment  of  his  report. 

CALIBRATION    OF    THE    MEASURING    RESERVOIR. 

Sump  No.  2,  constructed  of  concrete,  was 
approximately  rectangular  but  was  not  quite 
regular  in  form.  No  attempt  was  made  to 
base  computations  of  volume  on  its  linear 
dimensions,  but  a  scale  of  volumes  was  gradu- 
ated directly.  A  hook  gage  was  first  installed, 
the  hook  being  attached  to  a  vertical  rod,  to 
which  slow  motion  could  be  given.  An  index 
borne  by  the  rod  near  its  upper  end  followed  a 
smooth  surface  which  had  been  prepared  to 
receive  a  graduation  but  initially  was  un- 


THE    DISCHARGE-MEASURING    GATE   AND   ITS    RATING. 


259 


marked.  The  index  was  so  arranged  that  by 
pressure  it  could  be  made  to  indent  the  pre- 
pared surface,  thereby  producing  a  mark  of 
graduation.  Starting  with  just  enough  water 
in  the  sump  to  permit  the  use  of  the  hook,  a 
hue  of  graduation  was  marked.  Then  a  cubic 
foot  of  water  was  added,  the  hook  was  read- 
justed, and  a  second  line  was  marked.  By 
repeating  this  process  a  scale  was  given  to  the 
gage,  the  unit  of  the  scale  being  1  cubic  foot. 
The  added  units  of  water  were  measured  in  a 
wooden  bottle  made  in  the  form  of  a  cube 
with  the  opening  at  one  corner ;  and  the  volume 
of  the  bottle  was  so  adjusted  that  it  contained 
62.3  pounds  of  water.  The  capacity  of  the 
reservoir  was  about  38  cubic  feet. 

In  the  use  of  the  scale  thus  provided,  frac- 
tions of  a  cubic  foot  we/e  read  by  means  of  a 
small  free  scale  of  equal  parts  applied  obliquely 
to  the  space  between  two  lines  of  graduation. 

THE    OBSERVATIONS. 

The  record  of  an  observation  included  (1) 
the  width  of  gate  aperture,  (2)  the  time  inter- 
val, by  stop  watch,  during  which  the  water 
was  delivered  to  sump  No.  2,  (3)  two  readings 
of  the  hook-gage  scale,  and  their  difference, 
giving  the  volume  of  water  received  by  the 
sump,  and  (4)  two  gage  readings  of  water 
level  in  the  upper  tank,  one  just  before  and  one 
just  after  the  period  of  volume  measurement. 

The  quotient  of  the  volume  of  water  by  the 
time  in  seconds  gave  the  discharge  for  the 
indicated  width  of  gate  opening.  The  readings 
of  the  high-tank  water  level  gave  data  for  a 
correction  to  the  head,  resulting  in  a  small 
correction  to  the  computed  discharges. 

It  was  found  that  repeated  observations 
with  the  narrower  gate  openings  gave  results 
nearly  identical,  while  the  results  for  wider 
openings  showed  more  variation.  The  obser- 
vations with  wide  openings  were  accordingly 
multiplied  to  increase  the  precision  of  the 
averages.  The  variation  was  ascribed  to  pulsa- 
tions of  the  flowing  water  originating  in  the 
stilling  tank.  The  general  precision  of  the 


accepted  values  of  discharge,  listed  in  Table  2 
(p.  23),  is  indicated  by  an  average  probable 
error  of  ±0.2  per  cent.  The  largest  computed 
probable  error  is  ±  0.4  per  cent,  being  that  of 
the  discharge  for  a  gate  opening  of  6  inches. 

The  general  formula  for  discharge  through  a 
rectangular  orifice  when,  as  in  the  present  case, 
the  vertical  dimension  of  the  orifice  is  small  in 
relation  to  the  head,  is 


in  which  h  is  the  head,  measured  from  the 
middle  of  the  orifice,  Id  the  area  of  the  orifice, 
and  c  the  constant  of  discharge.  The  observa- 
tions give  the  following  values  of  c  for  different 
settings  of  the  gate: 


Width  of  gate 
opening. 

Value  of  c. 

Inches. 
1 
2 
3 
4 
5 
•  6 

Mean  

0.704 
.103 
.667 
.677 
.677 
.684 

.686 

For  a  "standard"  orifice  2  inches  square  and 
a  head  of  6  feet  Hamilton  Smith's  tables  1  give 
0.604  as  the  value  of  c.  The  inner  surface 
about  the  standard  orifice  is  vertical  and  plane, 
whereas  the  surfaces  about  the  orifice  of  our 
gate  were  oblique.  The  oblique  guiding  sur- 
faces served  to  increase  the  velocity  at  the 
orifice  and  thus  enlarge  the  constant  of  dis- 
charge. The  variation  of  the  constant  with 
width  of  opening  is  probably  connected  with 
the  fact  that  the  beveled  surfaces  were  not 
symmetrically  arranged  about  the  opening. 
On  three  sides  they  made  an  angle  of  45°  with 
the  vertical  plane  of  the  orifice,  but  the  edge 
of  the  slide,  constituting  the  fourth  side,  was 
beveled  at  a  smaller  angle.  The  constants  of 
the  gate  were  thus  affected  by  the  conditions 
of  its  setting  and  a  new  rating  would  be  neces- 
sary with  a  different  setting. 

i  Hydraulics,  p.  58. 


INDEX. 


•"••  Page. 

Accents,  notation  by 96 

Acknowledgments 9 

Adjustment  of  observations 55-95 

Airy,  Wilfred,  on  the  law  of  stream  traction 16,162 

Alluvium,  natural,  capacity  for 177-178 

Antiduues,  formation  and  movement  of 31-34, 243 

Apparatus,  descriptions  and  figures  of 1, 19, 257 

Arntzen,  "\VaIdemar,  work  of 9 

B. 

Bazalgette,  J.  W.,  on  flushing  sewers 216 

Bed ,  changes  in  roughness  of,  effect  of,  on  velocity 246 

characl  er  of,  effect  of,  on  flume  traction 206 

nature  of,  in  streams  and  in  flumes 15 

of  dcljris,  diagrammatic  longitudinal  section  of 57 

profiles  of 58 

stream,  composed  of  debris  grains,  ideal  profile  of 155 

contoured  plot  of 198 

Berlin,  laboratory  of  river  engineering  at 16 

Blackwell,  T.  E.,  observations  by,  on  velocity  competent  for  flume 

traction 216 

Blasius,  H.,  on  deposits  resembling  dunes 232 

on  rjiythmic  features  of  river  beds foot  note,  31 

Blue,  F.  K.,  experiments  by,  on  flume  traction 217-218 

Briggs,  Lyman  J.,  acknowledgments  to 9 

and  Campbell,  Arthur,  work  of,  on  viscosity  as  affected  by 

suspended  matter 228 

Brigham,  Eugene  C.,  and  Durham,  T.  C.,  work  of,  on  viscosity  as 

affected  by  suspended  matter 228 

Burgess,  J.  A.,  work  of 9 

C. 

Campbell,  Arthur,  and  Briggs,  L.  J.,  work  of,  on  viscosity  as 

affected  by  suspended  matter 228 

Capacity,  definition  of 10, 35 

for  flume  traction,  in  a  semicylindric  trough 214 

in  relation  to  discharge 209 

in  relation  to  fineness 210 

in  relation  to  slope 208 

of  mixed  grades 212 

table  of  adjusted  values  of 204-206 

in  relation  to  form  ratio 213 

lor  stream  traction,  in  relation  to  depth 164-168 

in  relation  to  discharge 137-149, 233-235 

in  relation  to  fineness 150-154, 235 

in  relation  to  form  ratio 124-136, 236 

in  relation  to  slope 96-120, 233 

in  relation  to  velocity 155-163, 193-195 

maximum 124, 130 

of  a  natural  alluvium 177, 180 

of  mixed  grades 113-115,169-185 

review  of  controls  of,  by  conditions 186-193 

table  of  values  of,  adjusted  in  relation  to  slope 75-87 

readjusted  in  relation  to  discharge 137-138 

readjusted  in  relation  to  fineness 151 

for  suspension 223-230 

Channels,  crooked,  experiments  with 196-198 

curved,  features  of 198,220-221 

form  of 10 

of  fixed  wjdth,  relation  of  capacity  to  slope  in 96-116 

of  similar  section,  relation  of  capacity  to  slope  in 116-120 

shaping  of,  by  natural  streams 221,222 

widths  of 22 

See  also  Bed  and  Form  ratio. 

Christy,  S.  B. ,  acknowledgments  to 9 

Competence,  definition  of 35 

influm   traction,  data  bearing  on _ 215-216 

simultaneous,  for  all  controls  of  capacity 187 

Competence  constants,  use  of  term 187 

Computation  sheet,  logarithmic,  figure  showing 95 

Contraction ,  local,  eflect  of,  on  velocity 245 


Contractor,  influence  of 57 

outfall,  description  of 25 

Controls  of  capacity,  review  of. 186-195 

Cornish,  Vaughan,  on  antidunes footnote,  32 

on  progressive  waves  in  rivers 244 

Cunningham,  Allan,  on  the  velocities  of  streams 155 

Current,  nature  of,  at  bends  in  channels. . . : 198, 220-221 

See  also  Velocity. 

D. 

Darcy,  H.,  and  Bazin,  F.  A.,  modification  of  Pitot-Darcy  gage  by.      262 

formulas  of,  for  velocities  in  conduits  and  rivers 193-194 

Deacon,  G.  F.,  experiments  by 16 

Di'-bris,  collective  movement  of. 30-34 

grades  of,  tables  of 21, 199 

mixed ,  causes  of  superior  mobility  of 178-179 

mixtures  of  grades  of,  evidence  from  experiments  with 113-1 15 

experiments  with 1 69-185 

table  of  observations  with 52-54 

movement  of  particles  of 155-156 

natural,  capacity  for  traction  of. 177-178 

table  of  observations  with 54 

source  and  sizes  of 21-22, 152 

table  of  observations  with , 36-54 

used  in  experiments,  plate  showing 22 

See  also  Fineness  and  Sand. 

Depth  of  water,  adjusted  values  of,  table  of. 89-93 

adjustment  of  observations  on 87-95 

gage  for  measuring 21 

in  relation  to  slope,  plot  of  observations  on 87 

in  unloaded  streams  in  flumes,  table  of 213 

method  of  measuring 25-26 

observations  on,  table  of. 38-54 

relation  of  capacity  to 164-168 

relation  of  velocity  to,  ideal  curves  showing 161 

table  of  values  of,  for  debris  of  grade  (C) ,  when  the  width  is  0.66 

loot  and  the  slope  is  1.0  per  cent 128 

variation  of,  as  related  to  variation  of  d  ischarge 165 

Dimensions  of  coefficients 64, 129, 139, 151, 186, 191 

Discharge,  change  of,  effect  of,  on  velocity 246 

competent,  table  of  experimental  data  on 70 

constant,  relation  of  capacity  to  depth  under 164 

relation  of  capacity  to  velocity  under 157 

control  of  constants  by 132-133 

definition  of. 35 

diversity  of,  rn  natural  streams 221-222 

gate  for  measuring,  description  and  rating  of 257-259 

measurement  of 22-23 

mode  of  controlling 20 

relation  of ,  to  « 66-67 

relation  of  capacity  to 137-149 

values  of,  corresponding  to  gate  readings 23 

variation  of,  in  relation  to  variation  of  depth 165 

Discharge  factor,  applicability  of,  to  natural  streams 2)3-235 

influence  of 10 

Dresden,  laboratory  of  river  engineering  at 16 

Dubuat-Nancay,  L.  G.,  experiments  of,  on  competent  velocity.  193,216 

Dunes,  definition  of 31 

formation  and  movement  of 31, 231-232 

longitudinal  section  illustrating 31 

In  relation  to  rhythm  of  current 242,244 

interval  between,  length  of 243 

Dupuit,  theory  of,  on  suspension footnote,  224 

Durham,  T.  C.,  and  Brigham,  Eugene  C.,  work  of,  on  viscosity  as 

affected  by  suspended  matter 228 

Duty,  constant  for  similar  streams 239 

definition  of 36, 74 

general  formulation  of 192 

In  experiments  of  F.  K.  Blue 217 

in  relation  to  discharge 144, 147 

in  relation  to  fineness 154 

in  relation  to  slope -      121 

values  of,  corresponding  to  adjusted  values  of  capacity,  table  of.  75-87 

261 


262 


INDEX. 


E.  Page. 

Efficiency,  definition  of 36 

general  formulation  of 192 

in  relation  to  discharge 144, 148 

In  relation  to  fineness 154 

in  relation  to  slope 121-123 

values  of,  corresponding  to  adjusted  values  of  capacity,  table  of.  75-87 

Efficiency  and  capacity,  table  comparing  parameters  in  functions 

Of 123 

Eger,  Dix,  and  Seifert  on  scale  of  a  model  of  Weser  River 237 

Energy,  effect  of  load  on 11, 225-227 

relation  of,  to  competent  slope 64 

Error,  probable,  for  stream  traction 56, 73-74,94, 113, 142-143, 151-153 

probable,  for  flume  traction 206 

Eshleman,  L.  E.,  work  of 9 

Experiments,  method  of  making 10,  22-26 

nature  and  scope  of 17-18 

results  of 240 

F. 

Fargue,  L.,  cited 197 

on  scale  of  model  river footnote,  237 

Feeding  of  debris,  suggested  apparatus  for 241 

influence  of,  on  capacity 56 

methods  of 23 

Fineness,  bulk,  definition  of 21, 35, 183 

bulk,  relation  of,  to  values  of  <r ,  table  showing 68 

competent 151 

constant  linear,  variations  of 153 

control  of  constants  by 133 

influence  of,  on  the  relation  of  capacity  to  form  ratio,  table  of 

quantities  illustrating 133 

in  natural  and  artificial  grades  of  debris,  range  and  distribu- 
tion of,  curve  illustrating 181 

linear,  definition  of 21, 35, 183 

mean,  changes  of,  in  natural  streams 221-222 

definition  and  measurement  of 182-184 

mixtures  of  diflerent,  experiments  with 169-185 

of  grades  of  debris 21, 199 

of  mixed  grades  and  their  components 180 

relation  of,  to  a 67-71 

relation  of  capacity  to 150-154 

for  natural  grades 180-182 

Fineness  factor,  applicability  of,  to  natural  streams 235-236 

influence  of 10 

Flow  of  water,  rhythm  in 242-244, 248-249 

Flume  traction,  definition  of 15 

discussion  of 199-218 

dune  action  in 202 

experiments  with,  apparatus  used  in 199-201) 

grades  of  de'bris  used  in,  table  of 199 

movement  of  de'bris  in 201-203 

mbvement  of  particles  in 200-201 

nature  of 11-12 

rough  surfaces  used  in  experiments  on,  plate  showing 200 

scope  of  experiments  on ..  18-19 

speeds  of  coarse  and  fine  de'bris  in,  table  of 200 

table  of  observations  on 202-203 

Form  factor,  influence  of 10 

Form  ratio,  computation  of 94 

definition  of 35-36 

discussion  of,  table  of  observations  for 51-52 

for  natural  streams 223 

influence  of 116-117 

of  unloaded  streams  in  flumes,  table  of 213 

optimum 134-136 

relation  of  capacity  to 124-136,190-192 

required  to  enable  a  given  discharge,  on  a  given  slope,  to 

transport  its  maximum  load,  table  of  values  of 135 

Form-ratio  factor,  applicability  of,  to  natural  streams 236 

Formulation  based  on  competence,  discussion  of 186-190 

Froude,  William,  on  proportionate  resistances  of  ships 237 


Grades  of  de'bris 21, 199 

See  alto  De'bris. 
Graphic  computation,  accuracy  of 95 

use  of 94-95 

Gue"rard,  Adolphe,  observations  on  the  Rhone  by 231 


H.  Page. 

Head  of  water,  determination  of 23 

Hearson,  T.  A.,  on  velocities  in  similar  streams 23" 

Hider,  Arthur,  on  dunes  in  Mississippi  River footnote,  31, 231 

on  intervals  between  dunes footnote,  243 

Hooker,  E.  H.,  cited 195  224 

Hopkins,  William,  on  the  law  of  stream  traction 16, 162 

Humphreys  and  Abbot  cited 229, 231 

I. 

Index  of  relative  variation,  nature  of . .  97-99 

synthetic,  nature  of 99 

Index  to  symbols 13-14 

Intake  influences,  nature  of 55 

Interpolation,  procedure  in 72-73 

Interpolation  formula,  selection  of 60-65 

J. 
Johnson ,  Willard  D . ,  acknowledgments  to 9 

K. 

Karlsruhe,  laboratory  of  river  engineering  at. ...  16 

L. 

Law,  Henry,  on  the  law  of  stream  traction 16,162 

Lechalas,  C.,  on  advance  of  dunes 232 

Lechalas,  formula  of _,  192-195 

Le  Conte,  J.  N.,  acknowledgments  to '. 9 

calibration  of  I'itot-Darcy  gage  by 252 

Leighton,M.  O.,  acknowledgments  to 9 

Leslie,  Sir  John,  on  the  law  of  stream  traction 16, 162 

Letters  and  symbols, index  to 13-14 

Levee,  natural,  formation  of 222 

Load,  definition  of 35 

effect  of,  on  the  stream's  energy 11,225-227 

effect  of  addition  of,  on  velocity , .  246-247 

effect  of  rhythm  on 53 

method  of  determining 24-25 

observations  on,  table  of 38-54 

partition  of. 230-233 

See  also  Capacity. 

Login,  T.,  experiments  by,  on  competent  velocity 163 

Loire  River,  observations  on 232 

subaqueous  dunes  of,  table  of  data  on 194 


McGee,  W  J,  originator  of  term  saltation 15 

McMath,R.  E.,  description  by,  of  the  movement  of  de'bris 156 

•  on  the  building  of  bars 232 

Manson,  Marsden,  on  suspended  load  of  Yuba  River 223 

Matthes,  Francois  E . ,  acknowledgments  to 9 

Mississippi  River,  dunes  in 31, 231-232 

suspended  load  of 229 

Mobility  of  debris,  causes  affecting 178-179 

Moving  field,  apparatus  for 27,249-250 

Movement  of  de'bris,  collective 30-34, 201 

modes  of 11,15 

Movement  of  par;  icles 26-30, 200-201 

Murphy,  E.  C.,on  the  movement  of  dunes  and  antidunes 32-33 

work  of 9 


Notation ..  13-14 


Observational  series,  definition  of 55 

Observations  on  load,  slope,  and  depth,  table  of 38-54 

on  competent  discharge,  table  of " 70 

on  competent  slope,  table  of 69 

See  also  Experiments. 
Outfall,  contracted,  effect  of  approach  to,  on  velocity 245 

effect  of  approach  to,  on  velocity 244-245 

Outfall  influences,  adjustments  for 56-57 

Overstrom,  G.  A., experiments  by,  on  capacity  of  launders.  210,216-217 
Owens,  John  S.,  experiments  by,  on  competent  velocity 163 

on  antidunes footnote,  32 


IKDEX. 


263 


P.  Page. 

Particles,  fineness  of 10,11 

modes  of  movement  of 1 1, 26-30, 200-201 

See  also  Fineness. 

Partiot,  H.  L.,  cited 194 

observation  by ,  in  the  Loire 232 

Pitot-Darcygago,  special  form  of,  description  of 251-252 

difficulties  in  the  use  of 26 

efficiency  of 255 

rating  of 252-256 

ratios  of  suction  at  one  opening  of,  to  pressure  at  the  other  ...      254 

variation  of  constant  of,  near  boundaries  of  current 254-255 

Plots,  logarithmic,  making  and  study  of 59-60 

Poiseuille,  J.  L.  M.,  theorem  established  by 155 

Power  function  and  its  logarithmic  locus 97-99 

Powless,  W.  II.,  observations  by 232 

Precision,  approximation  to,  for  stream  traction 56, 

73-74,94, 113, 142-143, 151-153 
approximation  to,  for  flume  traction 206 

R. 

Rankine,  W.  J.  M.,  formula  of,  for  resistance  of  a  wheel 201 

Reynolds ,  Osborne ,  on  the  flow  of  water!  hrough  tubes 242 

experiments  by,  with  one  liquid  flowing  over  another 248 

on  scales  of  model  tidal  basins 237 

Rhone  River,  observations  on  load  of 231 

Rhythm,  occurrence  and  influence  of. 58-^59 

problems  associated  with 241-250 

Richards,  E.  H.,  on  capacity  of  launders 210,216-217 

cited 226 

R  iver  engineering,  laboratories  of 16 

Rivers.    See  Streams,  natural. 

Rolling  of  particles,  features  of 26 

observations  on 200-201 

space  required  for 211 

S. 

Sainjon,  advance  of  dunes  formulated  by 232 

Saltation,  analysis  of 26-30 

comparison  of,  with  rolling 200 

definition  of 15 

zone  of,  curves  of  velocity  as  affected  by,  figure  showing 161 

Sand,  method  of  collecting 24 

mode  of  delivering 20, 23-24 

See  also  Ddbris. 

Sand  arrester,  description  of 20 

influence  of,  on  water  slope,  diagram  illustrating 57 

Scope  of  the  investigation 10 

Seddon,  J.  A.,  on  suspended  load  of  Mississippi  River 229 

Sensitiveness,  definition  of 188 

Settling  tank,  description  of 21 

Sewers,  experiments  on  traction  in 216 

Shearing,  interference  by  suspended  particle  with,  diagram  show- 
ing       226 

!),  relation  of,  to  discharge 66-67 

relation  of,  to  fineness 67-71 

to  width 67 

values  of,  adopted  for  Interpolation  formulas 72 

in  formulas  lor  mixed  grades  of  debris 169 

in  formulas  for  streams  of  similar  slope 11 7-119 

Sigma  formula,  use  of 96-97 

Sigma  function,  discussion  of. 99-100 

Sizes  of  debris 21, 199 

See  also  Debris. 

Sliding  of  particles,  infrequency  of 26 

observations  on 200-201 

Slope,  capacity  for  flume  traction  in  relation  to 203-206, 20S-209 

capacity  for  stream  traction  in  relation  to 55, 61-65, 96-123 

competent,  definition  of 35 

in  flume  traction 203 

observations  on 69 

relation  of,  to» 64-70,187 

definition  of 35 

effect  of  change  of,  on  velocity 245-246 

for  straight  and  crooked  channels,  table  comparing 197 

influence  of,  on  the  relation  of  capacity  to  form  ratio 132 

method  of  determining 21, 25 

observations  on,  adjustment  of 55-87 


Page. 

Slope,  observations  on,  in  crooked  channels 196-197 

observationsjon,  reduction  of 37 

table  of 38-54 

unit  of,  effect  of  changing 112-113 

Slope  and  mean  velocity,  relative  variation  of 158-159 

Slope  factor,  applicability  of,  to  natural  streams 233 

influence  of 10 

Smith,  Hamilton,  tables  of  discharge  by 259 

Speeds  of  particles  in  flume  traction,  table  of 200 

Streams,  natural,  application  of  laboratory  results  to 219-240 

natural,  kinds  of 219 

forms  of  channel  in 124, 222 

similar,  hypothesis  of 236-240 

Subsidence,  velocity  of 225-226 

Surfaces,  rough,  used  in  experiments  on  flume  traction 200, 2S6-208 

Suspension,  analysis  of 223-230 

definition  of 15 

forces  active  in,  diagram  of 224 

in  natural  streams 223-230 

relation  of,  to  flume  traction 200, 201 

to  traction  in  natural  streams 221-223 

Symbols,  index  to 13-14 

T. 

Tables,  list  of 5-6 

Terms,  definitions  of 35-36 

Thomson,  J.,  studies  by,  on  currents  of  streams 220 

Traction,  definition  and  classification  of 15 

flume,  modes  of 200-202 

observations  on... 202-203 

stream,  foreign  work  on 15-16 

modes  of 26-34 

observations  on 36-54 

See  also  Flume  traction. 

Trassportat ion,  modes  of 1 1,  IS,  26-34 

Trough,  forms  and  sizes  of 16 

semicylindric,  capacity  for  flume  traction  in 214 

width  of 22 

relation  of,  to  a 67 

with  glass  panels  and  sliding  screen 27 

with  local  contraction,  velocities  in 245 

Troughs  witb  bends,  plans  of 196 

U. 
Units  used 34 

University  of  California,  acknowledgments  to 0 

apparatus  on  campus  of,  plate  showing Frontispiece. 

V. 

Van  Orstrand,  C.  E. .acknowledgments  to 9 

Velocity,  competent,  experiments  on 69, 70, 162-163 

influence  of  suspended  load  on 225-230 

Influence  of  tractional  load  on 229-230 

maximum,  hypothesis  as  to  posit  ion  of 248 

mean,  and  slope,  relative  variation  of 158-159 

computation  of 94 

of  streams  with  and  without  tractional  load,  table  compar- 


ing. 


230 


methods  of  measuring 26 

relation  of  capacity  to 155-163 

vertical  curve  of,  as  modified  by  conditions 244-247 

Viscosity,  influence  of 226-229 

Voids,  percentage  of,  effect  of  mixed  grades  on 179 

Von  Wagner,  on  the  velocities  of  streams 155 

W. 

Water,  mode  of  handling 19 

rhythm  in  the  flow  of 242-244 

surface  of,  form  of  profile  of 57 

profiles  of,  showing  undulat  ions  associated  with  ant idunes.       33 

relation  ol  slope  of,  to  capacity 

rhythmic  fluctuations  of 58 

slope  of,  as  affected  by  feeding 56 

Water  circuit,  diagram  of • 

Water  supply,  qua! ity  of 

Whirls,  movement  of  sand  by 32 

Woodward,  R.  S., acknowledgments  to 

Yuba  River,  observationsion '. 223-224, 230-231 


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